ALGEBRAIZATIONS OF QUANTIFIER LOGICS, AN INTRODUCTORY OVERVIEW 12.1th version, January 1997.

István Németi

Abstract.

This work is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics as well as those propositional logics (like modal logics) in the semantics of which theories of relations play an essential rôle. This work has a survey character, too. The most frequently used algebras like cylindric–, relation–, polyadic–, and quasi–polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some directions we tried to give an overview of the most recent results and research trends, too.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Getting acquainted with the subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Decidability, relativized versions, non-square approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. Algebras for logics with equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Axiomatic approach, “abstract” cylindric algebras (CA’s), RA’s etc. . . . . . . . . . . . . . . . . . . . . . . . 20 Advanced topics in RCA theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5. Algebras for logics without equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Advanced topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6. Algebras for logics extending L_{ωω} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Algebras for infinitary logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

7. Other algebras of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 (1) Many-sorted cylindric algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49 (2) Partial cylindric algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (3) Algebras of sets of finite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 (4) Algebras of generalized finite sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 (5) Making function symbols of logics explicit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52 (6) Cylindric-like algebras for nonclassical quantifier logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 (7) The Rasiowa-Sikorski approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 (8) Connections with and approaches originating from computer science. . . . . . . . . . . . . . . . . . . .53 (9) Connections with category theoretical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 (10) Peano Arithmetic, Gödel’s incompleteness etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 (11) Boolean algebras with operators (BAO’s). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56 (12) Logics without variables (combinatory logic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (13) Connections between algebras of logic and “pure logic”; universal algebraic logic . . . . . . . . 57 8. Appendix 1: The traditional finitely axiomatizable approximations CA, RA etc. of our classes RCA, RRA etc. of algebras of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 9. Appendix 2: Universal Algebraic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Research supported by Hungarian National Foundation for Scientific Research grant No’s. T16448, T7255, T7567.

A drastically shorter version of this work appeared as [N91].

§1. Introduction

The subject of this paper could be called, perhaps, first (and higher) order algebraic logic, or perhaps trans–Boolean algebraic logic, or algebra of relations of various ranks. This theory of algebras of relations (of various ranks) goes back to 140 years, cf. e.g. [D1864]. For aspects of history not covered here cf. [HMTII], [TG87], Anellis–Houser [AH89], [JSL], Pratt [Pr92], Maddux [Ma91].

This paper was written for a variety of readers ranging from the beginner to the specialist. In particular, no familiarity with algebraic logic is presupposed, moreover, we tried to avoid relying on familiarity with logic itself as much as we could. The reader not familiar with logic should be able to understand the paper modulo ignoring certain remarks. He/she is advised to view the paper as a study of relations of various ranks, and as that of algebras whose elements are such relations. Familiarity with the basics of naïve set theory and some basic concepts of universal algebra should suffice. A pleasant, richly motivated introduction to the subject is found in Halmos [Ha85, pp. 206–215], which seems to be complementary to the present paper.^{1}

When discussing the algebraizations of logics, we treated the semantic aspects (models, meanings of formulas in models etc.) of logics as primary, and the syntactic aspects (provability etc.) as secondary. This does not represent value judgment, nor does it mean that the syntactic aspects will be ignored, but instead it has two purposes. (1) We had to choose a unifying principle making the paper coherent and easy to read for the nonspecialist. (2) This way the main bulk of the paper will be understandable and appreciable for readers not too familiar with logic, we hope.

In §§2–5 we gradually introduce, and carefully explain the most frequently used algebras of this area: Cylindric algebras, relation algebras, and (quasi-) polyadic algebras (not necessarily in this order). At the same time, but in less detail, we discuss other algebras suitable for algebraizing unusual quantifier logics, e.g. fragments^{2} of L_{ωω}, higher order logic, many-sorted logic etc. In §6 we begin to speed up, and discuss algebraic counterparts of infinitary logics, and other abstract model theoretic logics with an emphasis on polyadic algebras. §7 is a quick overview of other approaches, where the change comes either from the method of algebraization, or from the choice of logic to be algebraized (roughly speaking). This overview also touches upon connections with computer science, and category theoretic logic. §7 is also intended to be a continuation of the similar overview at the end of [HMTII]. §8 (Appendix 1) deals with the algebraic counterparts of the syntactic aspects of logic. (These are the very abstract cylindric (etc.) algebras which need not be isomorphic with any kind of concrete algebras of relations.) §9 (Appendix 2) contains some universal algebraic background. In the paper, there is a further item (describing a special topic, and) running across the various sections, beginning with Remark 2 in §3, and recurring in each subsequent section. This item is devoted to an area full of intriguing famous problems which might be rather rewarding to work on. The area is known as “finitizability problem” or “finitization” for short, and became quite active recently. (The beginner may safely omit the parts dealing with finitization, at a first reading. Similarly, the reader not interested in the proofs, or in some of them, may safely skip them.)

Finally we should point out that because of the introductory character of this paper, there are several central areas of trans–propositional algebraic logic which are not even mentioned here. Therefore, having read this paper, it is advisable to read [M91], leaf through [HMTII] and then perhaps [TG87]. Actually, [M91] can be read equally well before, after, or parallel with reading the present work. The important work [M91] was written after the first versions of the present work were completed, therefore [M91] is not referred to as often as it should.

From now on, we will simply write algebraic logic instead of trans–propositional algebraic logic.

1For completeness, we note that the up-to-date basic source book for “first–order (and related) algebraic logic” (i.e.

for the subject matter of this paper) is [HMTII] supplemented with [HMT] and [TG87]. ([HMTII] serves also as the standard reference monograph.)

2E.g. the positive fragment also called coherent fragment in category theoretic logic.

§2. Getting acquainted with the subject

The algebraization of classical propositional logic, yielding Boolean algebras (BA’s), was immensely successful. It is fairly well understood how to extend this algebraization to other propositional logics (Blok–Pigozzi [BP89]). These new algebraizations were also very successful. What happens then if we want to extend the original algebraization yielding BA’s to first–order logic?

What are BA’s? They are algebras of unary relations. I.e. the elements of a BA 𝔅 are unary relations and the operations of 𝔅 are the natural operations on unary relations e.g. intersection, complementation. The problem of extending this approach to predicate logics boils down to the problem of expanding the natural algebras of unary relations to natural algebras of relations of higher ranks, i.e. of relations in general. The reason for this is, roughly speaking, the fact that the basic building blocks of predicate logics are predicates, and the meanings of predicates can be relations of arbitrary ranks. Indeed, already in the middle of the last century, when De Morgan wanted to generalize algebras of propositional logic in the direction of what we would call today predicate logic, he turned to algebras of binary relations.^{3} That was probably the beginning of the quest for algebras of relations in general. Returning to this quest, the new algebras will, of course, have more operations than BA’s, since between relations in general there are more kinds of connections than between unary relations (e.g. one relation might be the converse, sometimes called inverse, of the other).

The framework for the quest for the natural algebras of relations is universal algebra. The reason for this is that universal algebra is the field which investigates classes of algebras in general, their interconnections, their fundamental properties etc. Therefore universal algebra can provide us for our search with a “map and a compass” to orient ourselves. There is a further good reason for using universal algebra. Namely, universal algebra is not only a unifying framework, but it also contains powerful theories. E.g. if we know in advance some general properties of the kinds of algebras we are going to investigate, then universal algebra can reward us with a powerful machinery for doing these investigations. Among the special classes of algebras concerning which universal algebra has powerful theories are the so called discriminator varieties and the arithmetical varieties (cf. the textbooks [BS81], [MMT], [W78]). At the same time, algebras originating from logic turn out to fall in one of these two categories, in most cases. More concretely, more than half of these algebras are in discriminator varieties and almost all are in arithmetical ones. Certainly, all the algebras studied in the present paper are in arithmetical varieties. Therefore, awareness of these recent parts of universal algebra can be rewarding in algebraic logic. In order to preserve the introductory character of this paper, we will not assume familiarity with these theories of universal algebra. From time to time we will point out places where these theories can be usefully applied but these parts can safely be ignored. For completeness, the part of this machinery most relevant to the present paper is recalled in Appendix 2 in a form streamlined for easy understandability.

Let us return to our task of moving from BA’s of unary relations to expanded BA’s of relations in general. What are the elements of a BA? They are sets of “points”. What will be the elements of the expanded new algebras? One thing about them seems to be certain, they will be sets of sequences. Why? Because relations in general are sets of sequences. These sequences may be just pairs if the relation is binary, they may be triples if the relation is ternary, or they may be longer — or more general kinds of sequences. (There is another consideration pointing in the direction of sequences. Namely, the semantics of quantifier logics is defined via satisfaction of formulas in models, which in turn is defined via evaluations of variables, and these evaluations are sequences. The meaning of a formula in a model is the set of those sequences which satisfy the formula in that model. So we arrive again at sets of sequences.) So, one thing is clear at this point, namely that the elements of our expanded BA’s of relations will be sets of sequences. Indeed, this applies to all known algebraizations of predicate logics or quantifier logics, e.g. for cylindric algebras, quasi-polyadic algebras, polyadic algebras, Craig’s algebras, modal-cylindric algebras, algebraic counterparts of nonclassical predicate logics, higher order logics, abstract model theoretic logics, see [HMTII, §5.6, pp. 263–271].

3De Morgan illustrated the need for expanding the algebras of unary relations (i.e. BA’s) to algebras of relations in general (the topic of the present paper) by saying that the scholastics, after two millennia of Aristotelian tradition, were still unable to prove that if a horse is an animal, then a horse’s tail is an animal’s tail. (“v_{0} is a tail of v_{1}” is a binary relation.)

At this point it might be useful to point out that the most obvious approach based on the above observation (that the elements of the algebra are sets of sequences) does not seem to work, at least not without some fine-tuning. So, what is the most obvious approach? Consider some set U; let <ωU denote the set of all finite sequences over U, and consider the BA P(^{<ω}U) (the powerset of <ωU conceived as a BA the standard way). Now if we are given any finitary relation, say, R ⊆ U ×U over U, then R ∈ P(^{<ω}U). So P(^{<ω}U) contains all relations over U independently of their ranks. Therefore it might be a candidate for being the universe of an algebra of relations. Before thinking about what the new, so called extra-Boolean operations on P(^{<ω}U) should be, let us have another look at its Boolean structure: If R is a binary relation, we would like to obtain its complement (U × U) ∖ R as a result of applying a Boolean operation to R. However, in our algebra P(^{<ω}U) the complement of R is not (U × U) ∖ R but something infinitely bigger, so this approach does not seem to work very smoothly without putting some extra effort into it. With further work it can be turned into a fruitful approach to algebraizing logic, see §7 (2–4) and the section containing Facts 2, 3 at the end of §4 herein; see also [HMTII,§5.6.(A survey).3, p. 265] and the references therein. The approach originates with Craig, but already the algebras in Quine [Q36] consist of sets of finite sequences.

§3. Binary relations

The above difficulty with P(^{<ω}U) motivates our concentrating first on the simplest nontrivial case, namely that of the algebras of binary relations (BRA’s). Actually, BRA’s will be strong enough to be called a truly first–order (as opposed to propositional) algebraic logic, namely [TG87, §5.3] shows that the logic captured by BRA’s is strong enough to serve as a vehicle for set theory and hence for ordinary metamathematics.

The \underline{full BRA over a set U} is defined to be the algebra

⟨𝔓(U × U), ∘,^{−1}⟩

where 𝔓(U × U) is the BA with universe P(U × U), R ∘ S is the usual composition of the binary relations^{4} R and S, i.e.

R ∘ S = {⟨a, b⟩ : ∃c(aRc and cSb)}

while R^{−1} = {⟨b, a⟩ : ⟨a, b⟩ ∈ R} is the converse of R. Throughout this paper, P(V) denotes the powerset of V, and 𝔓(V) denotes the BA with universe P(V) for any set V. By \underline{a full BRA} we understand the full BRA over some set U. By a BRA we understand a subalgebra of a direct product of full BRA’s, up to isomorphisms. Formally,

BRA = SP{⟨𝔓(U × U), ∘,^{−1}⟩ : U is a set}.

Throughout, we use abbreviations like BRA also for denoting the corresponding class itself, e.g. BRA also denotes the class of all BRA’s. Throughout, S and P are the operations of taking isomorphic copies of subalgebras and direct products respectively (as usual in universal algebra). We will see below that to any BRA 𝔄 there is an equivalence relation E (over some set) such that 𝔄 ≅ 𝔄^{+} ⊆ ⟨𝔓(E), ∘,^{−1}⟩ for some^{5} 𝔄^{+}. Whenever 𝔄 is embeddable into ⟨𝔓(E), ∘,^{−1}⟩ then we say that 𝔄 is representable by P(E) or is representable with top E. Note that if 𝔄 is representable this way, then 𝔄 is isomorphic to a Boolean algebra consisting of binary relations and closed under the operations ∘ and ^{−1}. So, every BRA is representable by an algebra consisting of binary relations.

4This is denoted by R|S in part of the literature, e.g. in [HMTII]. The reason for this is that in a large part of the literature, ∘ is reserved for the case when R and S are functions and is written backwards, i.e. what we denote by R ∘ S is denoted by S ∘ R.

The reason for this representability of BRA’s is the following. It is enough to see that products of full BRA’s are representable, since S obviously preserves representability. So let us consider products. Let us concentrate on the Boolean part first. It is easy to see that 𝔓(V) × 𝔓(W) ≅ 𝔓(V ∪^{.} W) where “≅” reads “is isomorphic with”, V and W are arbitrary sets and “∪^{.}” denotes disjoint union. (This is a well known easy fact of BA theory.) It is not hard to check that the proof smoothly generalizes from BA’s to BRA’s yielding

⟨𝔓(U × U), ∘,^{−1}⟩ × ⟨𝔓(V × V), ∘,^{−1}⟩ ≅ ⟨𝔓([U × U] ∪^{.} [V × V]), ∘,^{−1}⟩.

Notice here that [U × U] ∪^{.} [V × V] is an equivalence relation. So the product of any two full BRA’s is representable. The proof for arbitrary products goes exactly the same way. The proof also yields that if 𝔄 is represented with top element E = 1^{𝔄} and 𝔅 is represented with top R = 1^{𝔅} then 𝔄 × 𝔅 is representable with top E ∪^{.} R. The disjoint union of equivalence relations is again an equivalence relation. The just outlined simple fact, that direct products preserve representability, is used throughout all branches of algebraic logic, and accordingly it will be used here. So when defining new kinds of algebras of relations, we will first define the simplest version (e.g. the one with top element U × U × ⋯ × U), and then take all subalgebras of all direct products of these.

REMARK 1. Let us return to universal algebra as a unifying framework. If 𝔄 ⊆ ⟨𝔓(E), ∘,^{−1}⟩ as above, then the similarity type or signature of 𝔄 consists of the function symbols ∨, −, ∘,^{−1} where the first two are Boolean join and complementation. Homomorphisms, equations etc. are defined accordingly; e.g. homomorphisms should preserve all four operations, and (x ∨ y) ∘ z = (x ∘ z) ∨ (y ∘ z), (x ∨ y)^{−1} = x^{−1} ∨ y^{−1} are typical equations. The same convention applies to algebras of relations introduced later in the paper. The important thing to remember is that if 𝔄 = ⟨𝔅, f_{i}⟩_{i∈I} is a \underline{BA} 𝔅 expanded with additional \underline{o}perators f_{i} (a BAO from now on), then the algebraic language of 𝔄 is that of BA’s expanded with the extra-Boolean operation symbols f_{i} (i ∈ I). So in particular, a homomorphism h : 𝔄 ⟶ 𝔄 is a Boolean homomorphism preserving all the f_{i}’s. The same applies for equations, subalgebras, and other universal algebraic concepts. The literature of BAO’s is quite extensive, see e.g. [HMT, §2.7], Jónsson–Tarski [JT51], Jónsson [J91a], [J91], Henkin [H70], Goldblatt [G88], Sain [S84], Andréka–Givant–Németi [AGiN93] and §7 (10) herein.

Every variety of BAO’s is arithmetical by [MMT, Thm.4.43] or [BS81, Thm.II.12.5]. Hence the powerful theory of arithmetical varieties, cf. e.g. [MMT], [BS81] is applicable to practically all the algebras discussed in the present paper. Moreover, BRA’s generate a discriminator variety.^{6} This is proved in the proof of Thm.1 below.

Having a fresh look at our BRA’s with an abstract algebraic eye, we notice that they should be very familiar from the abstract algebraic literature. Namely, a BRA (3.1) 𝔄 consists of two well known algebraic structures, an involuted semigroup ⟨A, ∘,^{−1}⟩ and

a BA ⟨A, ∨, −⟩ sharing the same universe A. Further, the semigroup operations ∘ and ^{−1} distribute over the Boolean join ∨. Also ^{−1} is a Boolean isomorphism, while the smallest element 0 of the BA is a “zero” element of the semigroup i.e. 0 ∘ x = x ∘ 0 = 0.

5We note that ⟨𝔓(E), ∘,^{−1}⟩ would be a subalgebra of ⟨𝔓(U × U), ∘,^{−1}⟩ with E an equivalence relation on U if we would disregard the Boolean operations “−” and “1” (complementation and top element).

6Moreover, BRA’s, as well as all the other discriminator algebras studied in this paper, generate a doubly pointed discriminator variety in the sense of [BP89b]. This is useful e.g. because in any doubly pointed discriminator variety V, to any universally quantified formula φ there is an equation e_{φ} such that φ and e_{φ} are equivalent in the subdirectly irreducible members of V. Hence this is true for the varieties BRA, BRA^{0}, RA, CA_{n}, RPA_{α} etc. in this paper, see Theorem 14 below.

Property (3.1) defines a nice variety V_{+} containing BRA and is a reasonable starting point for an axiomatic study of the algebras of relations. Postulates in (3.1) already appear in De Morgan [D1864], and since then investigations of relation algebras have been carried on for almost 130 years.

THEOREM 1 (Tarski). BRA is a variety, i.e. is definable by a set of equations.

Preparations for the proof of Thm.1.: To illuminate the idea of the proof first we recall some standard material from Universal Algebra, and then we go through a few exercises. After that it will be easy and natural to see why and how the proof works.

Universal algebraic background 1:. Subdirect products of algebras, and subdirectly irreducible algebras are defined in practically every textbook on Universal Algebra, cf. e.g. Grätzer’s book [Gr69], or [BS81], [HMT], [MMT]. By a subdirect product we mean a subalgebra of a product such that the projections of the product restricted to the subalgebra remain surjective mappings. An algebra 𝔄 is subdirectly irreducible if it is not (isomorphic to) a subdirect product of algebras different from 𝔄. We note that the one-element algebra is not subdirectly irreducible. By Birkhoff’s classical theorem, every algebra is a subdirect product of some subdirectly irreducible ones. Therefore, the subdirectly irreducible algebras are often regarded as the basic building blocks of all the other algebras. In particular, when studying an algebra 𝔄, it is often enough to study its subdirectly irreducible building blocks.

For a class K of algebras, Sir(K) denotes the class of subdirectly irreducible members of K. Throughout, BA denotes the variety of all BA’s. For K = BA, Sir(BA) consists of the 2-element Boolean algebra only (up to isomorphisms). DEFINITION:

(i) We say that K has a discriminator term iff there is a term τ(x, y, z, u) in the language of K such that in every member of K we have

τ(x, y, z, u) = \{ \begin{matrix}z, if x = y, \\ u, otherwise. \end{matrix}

(ii) K is called a discriminator class iff Sir(K) has a discriminator term and K consists of subdirect products of members of Sir(K).

The term τ in (i) above is called a discriminator term. By a discriminator variety we understand a discriminator class which is a variety. 𝔄 is a discriminator algebra if {𝔄} is a discriminator class. Warning: Sometimes instead of the 4-ary τ, the ternary discriminator term t(x, y, z) = τ(x, y, z, x) is used. They are interdefinable, since τ(x, y, z, u) = t(t(x, y, u), t(x, y, z), z). Therefore, it does not matter which one is used.

EXERCISES 1.

(1) Prove that BA is a discriminator variety. Hint: use that BA is a variety and that the only member of Sir(BA) is the 2-element BA.

(2) Let 𝔅 = 𝔓({a, b}) where a ≠ b. Let c : B → B be defined by

c(x) = \{ \begin{matrix}∅, if x = ∅, \\ \{a, b\}, otherwise. \end{matrix}

Let 𝔄 = ⟨𝔅, c⟩.

– Prove that {𝔄} is a discriminator class. Hint: Prove that 𝔄 is simple, hence subdirectly irreducible. Then the discriminator term τ need to work on 𝔄. Write up τ from c(x).

– Try to prove that SP{𝔄} is a discriminator class, too. (If this is too hard, it can be skipped. But give it a try.) Hint: Show that Sir(SP{𝔄}) = S{𝔄}.

(3) Let 𝔄 = ⟨𝔓(2 × 2), ∘⟩ where 2 = {0, 1}. Prove that {𝔄} is a discriminator class. Hint: consider the term c(x) = 1 ∘ x ∘ 1. Prove that 𝔄 is simple, hence subdirectly irreducible, too. Write up the discriminator term from c (𝔄 has only four atoms, so a drawing of 𝔄 might help).

(4) Prove that BRA is a discriminator class. Hint: First prove that Sir(BRA) = S{⟨𝔓(U × U), ∘,^{−1}⟩ : U is a nonempty set}, by using Sir(SPK) ⊆ SK. Let the term c(x) be 1 ∘ x ∘ 1. Prove that in Sir(BRA),

c(x) = \{ \begin{matrix}0, if x = 0, \\ 1, else. \end{matrix}

Use c(x) for writing up a discriminator term.

(5) (This can be skipped.) Prove that BRA is contained in a discriminator variety. Hint: UpK denotes the class of algebras isomorphic to ultraproducts of members of K. First prove that K_{1} = SUpSir(K) consists of subdirectly irreducible algebras and is a discriminator class (here SUpK is the smallest class containing K which is axiomatizable by universally quantified formulas). Infer that SPK_{1} is a discriminator class containing K. Use Appendix 2 to conclude that SPK_{1} is a variety.

(6) Find an algebra 𝔄 which is not contained in any discriminator class! Hint: Let 𝔄 = ⟨{0, 1, 2}, suc⟩ where suc(n) = min{n + 1, 2}. Then 𝔄 is subdirectly irreducible but not simple. But the existence of a discriminator term τ on 𝔄 would force 𝔄 to be simple.

Now we are ready to turn to the proof of Thm.1.

By Exercise (4) (for short, Ex.4) we know that BRA is a discriminator class.

Claim 1.1. Sir(BRA) = S{⟨𝔓(U × U), ∘,^{−1}⟩ : U is a nonempty set} is closed under taking ultraproducts.

The equality part was proved in the proof of Ex.4. Hence it is sufficient to prove that Sir(BRA) is closed under ultraproducts. An ultraproduct of full BRA’s on some sets U_{i} is isomorphic to the full BRA on the ultraproduct of the U_{i}’s, namely if F is an ultrafilter on I, and 𝔉(U) denotes ⟨𝔓(U × U), ∘,^{−1}⟩, then Π𝔉(U_{i})/F ≅ 𝔉(ΠU_{i}/F), and the isomorphism h is given by a/F ↦ {⟨u/F, v/F ⟩ : {i ∈ I : ⟨u_{i}, v_{i}⟩ ∈ a_{i}} ∈ F }. The reader is invited to check that h is indeed an isomorphism. Thus K_{1} = {⟨𝔓(U × U), ∘,^{−1}⟩ : U is a nonempty set} is closed under ultraproducts. This implies that SK_{1} is also closed under ultraproducts. With this, Claim 1.1 is proved.

Thm.9.2 fully proved in Appendix 2 herein says that HSPK = SPUpK for any K having a discriminator term. Since Sir(BRA) does have such a term (cf. Ex.4), Thm.9.2 applies to it. Then by Claim 1.1, HSPSir(BRA) = SPUpSir(BRA) = SPSir(BRA) = BRA, by Thm.9.2, Claim 1.1 and the definition of BRA respectively. Since the HSP closure of any class is equationally axiomatizable, this proves Thm.1. QED(Thm.1)

Theorem 1 indicates that BRA is indeed a promising start for developing a nice algebraization of (at least a part of) first-order logic, or to put it more plainly, for developing an algebraic theory of relations. After Theorem 1, the question comes up naturally whether we can strengthen the postulates in (3.1) to obtain a finite set Σ of equations describing the variety BRA, i.e. such that BRA = Mod(Σ) would be the case.^{7} The answer is THEOREM 2 (Monk). BRA is not finitely axiomatizable, i.e. for no finite set Σ of first-order formulas is BRA = Mod(Σ).

7Mod(Σ) is the class of algebras in which Σ is valid.

The idea of one possible proof is explained in Remark 5.2 in §4 herein. This idea is based on the proof of Thm 5.1 which is the reason why it is postponed to that part of the paper. See Monk [M64], and also [HMTII, 5.1.57, 4.1.3] for the original proof of Theorem 2 (in slightly different settings).

Theorem 2 was strengthened by Andréka, Jónsson, Maddux and Venema. Maddux [Ma89] proved that the set of equations containing only one variable and valid in BRA is not finitely axiomatizable. Jónsson [J91a] proved that no set Σ of universally quantified formulas containing only finitely many variables can axiomatize BRA. Andréka proved the following two results. (1) Recall from Remark 1 that the basic operations of BRA are ∨, −, ∘,^{−1}.^{8} Then, any set Σ of equations axiomatizing BRA (or its expansion RRA to be introduced soon) has the following “complexity” property. For any k ∈ ω, there are infinitely many formulas φ ∈ Σ such that all the basic operations of BRA, with the exception of “^{−1}”, occur in φ and φ contains more than k distinct variables. This does not extend to the operation ^{−1} (and in the RRA case, to any of “^{−1}” or the constant Id), cf. [A91, Remark 9, pp. 83–84],[AN93], [A90]. Some of her further results in this “complexity” direction will be mentioned around Theorems 4, 5.1, 11 and in a footnote preceding Theorem 14. The other result is (2) no subreduct of BRA containing ∨ and ∘ among its operations can be axiomatized by a set of universally quantified formulas containing finitely many variables, or by a finite set of first-order formulas ([A89a]); where by a subreduct of a class K we mean the class of subalgebras of a reduct of K (to some language). Cf. also [A91, §III, pp.72–86]. For further improvements of Theorem 2 see also the remarks below Problem 2, Theorem 2.1, below Theorems 4, 5.1, below Remark 5.2 and second footnote of §6 herein.

Venema improved Andréka’s result in (1) above in a special direction as follows: Using a construction from Hodkinson [Ho96], Venema [V96] showed that no set of the so–called Sahlqvist formulas will axiomatize BRA. Sahlqvist formulas generalize the notion of positive equations (i.e. equations containing no complementation).

It turns out in the fundamental book [TG87] that Theorem 2 above and its improvements are essential for applications in finitizing metamathematics.

As a consolation for Theorem 2, by Monk [M69], one can obtain an example of a recursive (i.e. decidable) infinite set Σ of equations characterizing the variety BRA. Lyndon [Ly56] outlines another recipe for obtaining a different such Σ which may work for BRA. Hirsch–Hodkinson [HH96] also contains such a set Σ. However, the structures of these Σ’s are rather involved. Cf. [HMTII, pp. 112–119] for an overview. In this connection, we note that the following is still one of the most important open problems of algebraic logic:

PROBLEM 1. Find \underline{simple}, mathematically transparent, decidable sets Σ of equations characterizing BRA. (A solution for this problem has to be considerably simpler than, or at least markedly different from the Σ’s discussed above.)

Compare [HMTII, Problem 4.1, p. 179], Henkin–Monk [HM74, Probl.5] [M77, p. 85_{3}], [TG87, p. 240_{16−13}], [AGN77, p. 21], [W88], Venema [V90], [V89, Def.1.4.1, Thms 1.4.16, 3.3.5], Simon [Si90].

One can get very far in doing algebraic logic (for quantifier or predicate logics) via BRA’s. If we want to investigate nonclassical quantifier logics, we can replace the Boolean reduct 𝔅 of 𝔄 = ⟨𝔅, ∘,^{−1}⟩ ∈ BRA with the algebras (e.g. Heyting algebras) corresponding to the propositional version of the nonclassical logic in question. By Andréka’s above quoted result (2), if 𝔅 is (an expansion of, like Heyting algebras) a distributive lattice, then Theorem 2 carries over, cf. Theorem 2.1 below. We will return to this direction later. At a first reading, some parts of Remark 2 below might be too technical for the nonspecialist. They may be skipped safely, but it is advisable to come back to them, sometime.

Throughout, ω is the set of natural numbers, and it coincides with the least infinite ordinal.

8Other irredundant systems, e.g. ∧, −, ∘,^{−1} would suit our purposes just as well.

REMARK 2 (\underline{Finitization}). The efforts of trying to get rid of Theorem 2 became known as finitizability investigations. Why would we want to get rid of Theorem 2? BA’s of unary relations, i.e. SP{𝔓(U) : U is a set} admit a nice finite axiomatization. Theorem 2 says that the same is not possible for BRA’s of binary relations. Since we are in the middle of the search for the right notion of algebras of (not only unary) relations, the question naturally comes up whether Theorem 2 was perhaps only a consequence of an unfortunate choice of the basic operations ∘ and ^{−1} of BRA’s.^{9}

Apparently, it is hard to get rid of Theorem 2 by moving to reducts of BRA’s (cf. Andréka’s result mentioned below Theorem 2, Schein [Sc89], Andréka [A89, A89a]). It is a second central open problem^{10} of algebraic logic to find out whether we can get rid of Theorem 2 by moving in the direction opposite to taking reducts. This would mean to expand BRA’s by new, “set-theoretically defined” operations on relations such that the new class would become finitely axiomatizable. This quest is motivated by the fact that there are many known examples for a non-finitely axiomatizable class K of structures such that an expansion K^{+} of K is finitely axiomatizable. An example for such a K and K^{+} are the elementary classes generated by the structures ⟨ω, 0, succ⟩ and ⟨ω, 0, succ, ≤⟩ respectively. Another example is set theory. ZF is not finitely axiomatizable but its conservative extension known as Gödel–Bernays set theory is such.^{11} A drawback of these examples is that representability of members of K^{+} (as some kinds of algebras of relations) do not enter the picture. To alleviate this, we quote an example due to Bredikhin: Subalgebras of ⟨P(U × U), ∘,^{−1}⟩ for all U form a non-finitely axiomatizable quasivariety while substructures of their expansions ⟨P(U × U), ∘,^{−1}, ⊆, dom⟩ where dom(R) = Id ↾ Domain(R) = {⟨a, a⟩ : (∃b)(⟨a, b⟩ ∈ R)} form a finitely axiomatizable universal Horn class (Bredikhin [Br77], [Br77a]).^{12} (Both classes are understood up to isomorphisms, of course.)

The problem was raised in various forms by Jónsson, Henkin–Monk [HM74, Problem 1], Monk [M70, p. 20], Tarski–Givant [TG87, lines 9–11 of p. 62] (the first sentence of the paragraph preceding 3.5.(ix)), and the first page of [TG87, §3.5 (p. 56)], whether (3.2) one can add to the operations of BRA finitely many new set-theoretically defined operations

f_{1}, …, f_{n} on relations such that K^{+} = {⟨𝔓(U × U), ∘,^{−1}, f_{1}, …, f_{n}⟩ : U is a set} would generate a finitely axiomatizable variety.

A perhaps even more important version of the problem asks whether (3.2)^{+} statement (3.2) is true in such a strong form that for K^{+} in it, its closure SPK^{+} is a variety,

or at least is a finitely axiomatizable class.

What does it mean in (3.2) that f_{i} should be set-theoretically defined? The intuitive idea is that it should be somehow “concretely specified” like the operations of BRA. Namely, in the latter e.g. R∩S or R∘S are defined by set-theoretical means in a very strong sense, which entails, among other things, that the results of these operations depend only on the choice of R and S, independently of the choice of the algebra in which we execute them. So in a sense, these operations are “concrete” (as opposed to the abstract algebraic operations of fields for example). The most generally accepted precise version of this condition says that f_{i}, i ≤ n, should be invariant under the permutations of U. This means that whenever p : U ↣ → U is a permutation then p(f_{i}(R)) = f_{i}(p(R)) where^{13} p(R) is defined the natural way for R ⊆ U × U (p(R) = {⟨p(a), p(b)⟩ : ⟨a, b⟩ ∈ R}). The following is still open.

9Henkin pointed out during the 1987 Algebraic Logic conference in Asilomar that a positive solution for this problem would imply positive results for pure (non-algebraic) logic too; see Sain [S87a, §4], largely based on Henkin’s suggestions. According to Henkin, on his own part, this was the main reason for including this “goal” as Problem 1 in [HM74].

10This is not really a single problem; it is rather a large circle of problems motivated (and “punctuated”) by many deep results. In a sense it is a rich theory built around a deep, open, almost philosophical question.

11Fact 1 below is a third, rather general motivating example. Actually Fact 1 seems to imply that what we want to do is always possible. The only question is whether we can do it in such a way that the new operations remain “representable as natural set-theoretic operations on relations (like ∩, ^{−1}, and ∘ were)”. It is this latter (vague)

requirement which was referred to above as “set-theoretically definable”.

12Bredikhin’s dom(x) is the same as Craig’s T_{[1/0]}(x) to be discussed in §6 here.

PROBLEM 2. Is (3.2)^{+} above, or if not then at least (3.2) true, when “set–theoretically defined” means permutation invariant in the above sense?

The problem in this form was raised among others by Bjarni Jónsson sometime before 1983; see e.g. [S87, S87a], the part of [Bi89] coming after Corollary 10, and the parts of [TG87] indicated just before formulating (3.2) above. Biró [Bi89] proved that f_{1}, …, f_{n} cannot all be “first–order definable” in a certain sense of the term we will recall only vaguely here^{14}. In terms of “algebraizing logic”, f_{i} is “first–order definable” is equivalent to saying that f_{i} is the algebraic counterpart of a derived logical connective of first–order logic L_{ωω} (just as meet ∩ is the algebraic counterpart of the logical connective conjunction ∧). A typical example for a first–order definable operation is f which leaves the relation R fixed iff R has exactly 1000 elements and otherwise sends R to 0. A typical counterexample (i.e. non first–order definable) is obtained from this f by replacing 1000 in its definition with a fixed infinite cardinal. We will give more examples soon. Andréka has a proof–outline for an improvement of Biró’s result, where “first–order definable” is replaced with “either first–order definable, or generalized quantifier or a modality” (where f_{i} is a modality if it distributes over ∨), cf. the second footnote in §6 for more concrete formulation. However, Andréka checked the details for algebras of n–ary relations (RCA_{n}’s) to be discussed in §4 below, but not for the presently discussed BRA’s designed for binary relations only. Sain [S87a] contains a positive solution but for a different kind of algebra of relations to be discussed later. [Bi89] and independently [Ma89b] showed that the requirement of permutation invariance is essential (i.e. without this requirement there is a solution but the logical counterpart of this solution does not satisfy the axioms of abstract model theory i.e. those of general theory of logics).

Some possible choices of the f_{i}’s are: the identity relation Id on U as a new constant (distinguished element), transitive closure trc(R) = R ∪ (R ∘ R) ∪ (R ∘ R ∘ R) ∪ …, and a choice function picking out a singleton {⟨a, b⟩} ⊆ R from R, i.e. ch(R) ⊆ R such that |ch(R)| = min(1, |R|). Of these, Id is first-order definable, trc is not, but both are permutation invariant; ch is not permutation invariant (hence it is not allowed in Problem 2). Actually [Bi89], [Ma89b, Thm’s 8, 9] showed that adding ch makes BRA finitely axiomatizable. A further example for a permutation invariant but not first–order definable operation is fin(R) = “∅ if R is finite, else R”. (This operation will be relevant in §6.) Adding all the permutation invariant ones of these four operations to BRA does not make the class finitely axiomatizable.^{15}

Throughout, Rd is taken to be the operation of forming appropriate reducts. In particular, K_{1} = RdK_{2} means that we can identify the operation symbols of K_{1} with some of those of K_{2} such that after forgetting the rest of the operations of K_{2}, the two classes become identical, up to isomorphisms. (Beginning with §4 below, we will use Rd in a slightly more general sense, namely we will allow the operation symbols of K_{1} to correspond to term functions, i.e. derived operations of K_{2} instead of only operation symbols of K_{2}. However, in the present section the narrower definition suffices.) K_{1} = SRdK_{2} is defined in the same fashion. Now, (3.2) above implies that BRA = SPRd(K^{+}) for K^{+} as described in (3.2). The following is a variant of (3.2)^{+}:^{16} (3.3) There are set-theoretically defined operations f_{1}, …, f_{n} on binary relations, such that

13In §7(9) herein we will describe a category theoretic (functorial) version of this condition excluding the operation f(R)=“∅ if ∅ ∈ U else R”. According to the present (non-categorical) definition, this f is permutation invariant, but from the point of view of algebraizing arbitrary quantifier logics in the abstract model theoretic sense, f violates one of the axioms of abstract model theory; cf. Barwise–Feferman [BF85]. Cf. §7(9) herein.

14The definition can be found in Jónsson [J91a] or Biró [Bi89], but using cylindric algebras to be introduced later, basically, an operation on relations is “first–order definable” iff it is term definable in cylindric algebras (of ω-ary relations, i.e. in RCA_{ω}). We note that cylindric algebras are basically expansions of BRA’s as shown e.g. in [HMTII, §5.3], and sketched in §4 below.

15The case of {trc, Id} is in [Ma89b, Thm.3], but see also [AGN87], Ng [N84], [N81], and [M77, Prob.7] for investigations of trc. Further, Andréka proved that (i) “RRA + trc” is not finitely axiomatizable over RRA, and (ii) no expansion of “BRA + trc” is finitely axiomatizable. So if in (3.2) we would replace “∘,^{−1}” with “∘,^{−1}, trc” then the answer to all parts of Problem 2 would be NO, even if the condition “set-theoretically definable” were completely omitted.

K^{+} = S{⟨𝔓(E), f_{1}, …, f_{n}⟩ : E is an equivalence relation}

\underline{is} a finitely axiomatizable \underline{variety} and BRA = SRd(K^{+}). The main difference between (3.3) and (3.2)^{+} is the replacement of U × U in (3.2)^{+} by an

equivalence relation E; without this change we cannot expect K^{+} to be a variety (closure under P is the problem). This U × U ↦ E substitution was used to the same effect immediately below the definition of BRA.

Variants of Problem 2 ask if (3.3) is true but again in formulating these problems, one has to make the adjective “set-theoretically defined” concrete and precise.

The condition K^{+} = SK^{+} in (3.3) is important because, by [HMTII, 3.2.5], there is a finitely axiomatizable K^{+} such that BRA = SK^{+}. (Cf. also Jónsson–Tarski [JT51, Thm.4.31].) This also yields a kind of a characterization of the equations valid in BRA, namely all the equations derivable from the single formula axiomatizing K^{+} in [HMTII, 3.2.5]. This illustrates what kinds of answers are no longer acceptable as solutions for Problem 1, making the vague adjectives like “simple”, “transparent” in the formulation, hopefully, more tangible.

In (3.2) and (3.3) above (hence in Problem 2), the condition saying that f_{1}, …, f_{n} should be “settheoretically defined” (whatever concrete meaning the latter will receive later) is very important, because without this condition a version of the problem seems to receive a trivial solution by a result of Kleene from 1951 improved by Craig and Vaught [CV58]. Namely:

FACT 1. For any quasivariety K axiomatizable by a recursively enumerable set Σ of first-order formulas, there is a finitely axiomatizable K^{+} = SK^{+} such that K = SRd(K^{+}).

Proof: We will show how Fact 1 follows from Thm.2.1 of [CV58], which says that if Mod(Γ) consists of infinite models and Γ is recursively enumerable, then Mod(Γ) = Rd(K^{+}) for some finitely axiomatizable class K^{+}. Let Σ be as above. Form Σ_{∞} by adding to Σ postulates stating that “if there are at least two elements, there are at least n different elements” for each n ∈ ω. Then K = SMod(Σ_{∞}) and each model of Σ_{∞} is infinite (except for the one-element trivial one which may be safely ignored). By Thm.2.1 of [CV58], there is a finitely axiomatizable K^{+} with Mod(Σ_{∞}) = Rd(K^{+}). But then K = SRd(K^{+}). Finally, the predicates in K^{+} are easily coded by their characteristic functions, so K^{+} can be transformed into a class of algebras. By introducing Skolem functions, we can achieve K^{+} = SK^{+} (note that only finitely many Skolem functions are needed because only finitely many axioms are needed for describing K^{+}). Fact 1 has been proved.

We conjecture that Fact 1 can be improved to conclude that K^{+} is a finitely axiomatizable quasivariety.

Fact 1 immediately yields that version of (3.3) from which the condition “set-theoretically defined” is omitted, and in which “variety” is replaced by “universally axiomatizable class”.

Sain [S87] proved that version of (3.3) in which “set-theoretically defined” is made precise as follows: f_{i} is set-theoretically defined iff we can write down a concrete set-theoretical formula φ_{i}(x_{1}, …, x_{n}, y) containing bounded quantifiers only such that for any binary relations X_{1}, …, X_{n}, Y we have [f_{i}(X_{1}, …, X_{n}) = Y ⟺ φ_{i}(X_{1}, …, X_{n}, Y)]. This immediately ensures that f_{i}(R) depends only on R and not on the choice of the algebra in which f_{i}(R) is computed.

16To avoid misunderstandings, we are not saying that (3.3) would be true; we only formulate it as a “statement” to be discussed later.

We note that, in a sense, Sain’s result improves Maddux’s and Biró’s choice function ch because ch is not definable in general in set theory, especially not by an absolute^{17} formula like φ_{i} above. All these leave Problem 2 open since there “set-theoretically defined” is interpreted as meaning invariant under permutation.

In passing we note that Tarski and Givant [TG87, p. 62] raise a perhaps easier version of Problem 2. Namely, they ask for a finitely axiomatizable variety K_{0} containing K^{+} of (3.2) such that BRA = SRd(K_{0}). Here, the point is that not all members of K_{0} have to be “representable” by SPK^{+} or even by HSPK^{+} (but still K^{+} ⊆ K_{0} is required and f_{i} of K^{+} has to be permutation invariant)^{18}. This problem, at least in principle, is indeed easier than Problem 2, by the following example. Andréka proved that if Ω = {∨, ∘} and Γ = Ω ∪ {∧} then SRd_{Ω}BRA and SRd_{Γ}BRA are not finitely axiomatizable, but there is a finitely axiomatizable variety K_{0} ⊇ SRd_{Γ}BRA with SRd_{Ω}BRA = SRd_{Ω}(K_{0}). Here Rd_{Ω} denotes taking reducts to the similarity type Ω. This “easier” version of Problem 2 is also open. The following improvement of Theorem 2 is relevant to the finitization problem.

In the next theorem we use the class RRA of Representable Relation algebras. This class will be introduced very soon^{19}.

THEOREM 2.1 (H. Andréka – Sz. Mikulás). Let V be a variety term definitionally equivalent with RRA. Let V_{0} be a subreduct of V. That is, V_{0} = SRd_{Ω}V for some set Ω of operation symbols. Assume ∩, ∪, ∘ are among the operations of V_{0}. Then V_{0} is not finitely axiomatizable.

Proof (Mikulás): In the proof of Theorem 4 in [A89b] a sequence ⟨𝔄_{n} : n ∈ ω⟩ of non–representable relation algebras is defined such that its ultraproduct Π_{n∈ω}𝔄_{n}/F ∈ RRA. Further it is proved that the {∩, ∪, ∘}–reduct of 𝔄_{n} is not embeddable into any reduct of any RRA. We may assume that the operations of V_{0} form Ω = {∩, ∪, ∘} ∪ {τ_{1}, … τ_{k}}, where τ_{i} is a term–function of RRA for each i ≤ k. Then ℜ𝔡_{Ω}𝔄 ∉ SRd_{Ω}RRA because of the above quoted property of 𝔄_{n}. But then ℜ𝔡_{Ω} (Π_{n∈ω}𝔄_{n}/F) ∈ Rd_{Ω}RRA proves that the complement of V_{0} = SRd_{Ω}RRA is not closed under ultraproducts.

Intuitively, Theorem 2.1 says that no expansion of ∩, ∪, ∘ with operations term definable in RRA yields a finitely axiomatizable notion of set–algebras. We conjecture that here RRA–term definable can be replaced with “first–order definable” in the sense of [Bi89], Jónsson [J91a].

We will return to the finitization problem later, when we will have a richer theory at our hands suitable for developing a deeper understanding of the problem. Cf. e.g. the last 2 pages of §4 especially Problems 2.1, 2.2. Cf. also the part beginning with Thm.16 and containing item (5.1). Cf. also Problem 3 in §7(9). \underline{END of Remark 2}

The natural logical counterpart of BRA’s is classical first-order logic restricted to three individual variables v_{0}, v_{1}, v_{2} and without equality. As shown in [TG87, §5.3], this system is an adequate framework for building up set theory and hence metamathematics in it. One can illustrate most of the main results, ideas and problems of algebraic logic by using only BRA’s. (An illustration of this is the result just quoted from [TG87, §5.3.]) We do not know how far BRA’s can be simplified without losing this feature. In this connection, let the class BSR of Boolean semigroups of relations be defined as

17The φ_{i}’s are known to be absolute in a very strong sense used in set theory, see Barwise [B75]. Note, however, that Sain’s result does not give us a permutation invariant operation. Actually, the two properties are independent:

there are absolute operations which are not permutation invariant, and also permutation invariant operations which are not absolute. Sain has a permutation invariant result too, but for that, relations of higher ranks are also needed as elements of the algebra. Therefore this result will be stated as Theorem 16 in §6.

18Their wording of the problem is different from the present algebraic one, since they formulate the problem in its logical form. (The same remark applies to our quoting [TG87] in connection with Problem 2 above.) In logical form, the presently discussed easier question does not require the expanded logic L^{+} corresponding to K^{+} to be complete, but L^{+} still has to be sound. (And of course, L^{+} has to be complete w.r.t. the old formulas.) Strangely enough, this soundness requirement (together with permutation invariance) seems to keep the problem on the tough side. Maddux [Ma89b] proved that adding {Id, trc} does not solve this easier problem either.

19RRA is obtained from BRA by adding the identity relation Id_{U} = {⟨a, a⟩ : a ∈ U} as a distinguished constant.

So, a typical RRA is a subalgebra of ⟨𝔓(U × U), ∘,^{⌣}, Id_{U} ⟩ for some set U.

BSR = SP {⟨𝔓(U × U), ∘⟩ : U is a set} .

So we require only one extra-Boolean operation “∘”.^{20} The question is, how far BSR could replace BRA as the simplest, “introductory” example of Tarskian algebraic logic. We conjecture that the answer will be “very far”. We know that BSR is a discriminator variety, and is not finitely axiomatizable. Thus Theorems 1, 2 remain true if BRA is replaced with BSR in them. Further, the equational theory of BSR is undecidable. We conjecture that, following the lines of [TG87, §5.3], set theory can be built up in BSR instead of BRA with basically the same positive properties (e.g. finitely many axioms) as the present version [TG87] has.^{21} It would be nice to know if this conjecture is true, and, more generally, to see a variant of algebraic logic elaborated on the basis of BSR. We do not know what natural fragment of first-order logic with three variables corresponds to BSR (if any). It certainly is difficult to simulate substitution of individual variables using only ∘. The converse operation, ^{−1}, is the algebraic counterpart of substitution because, intuitively, R(v_{0}, v_{1})^{−1} = R(v_{1}, v_{0}). One can simulate quantification by ∘, and it is easily seen that ∘ is stronger than quantification but without ^{−1} it is not clear exactly how much stronger. Curiously enough, these issues are better understood in the case of cylindric algebras to be discussed later.

BRA’s also play an important rôle in theoretical computer science (cf. e.g. Hennessey [H80], Bednarek–Ulam [BU77], Imieliński–Lipski [IL84], Berghammer–Zierer [BZ86], Hoare–Jifeng [HJ86], [SS89], van Benthem [vB90a, vB90], and §7 (7) herein). Jónsson [J89], [J90] call a finitely axiomatizable variety BRA^{0} approximating BRA Program Specification Algebras. Here, by saying that the finitely axiomatizable BRA^{0} approximates the not finitely axiomatizable BRA we mean that BRA^{0} ⊇ BRA and BRA^{0} is fairly close to^{22} BRA.

If we want to algebraize first-order logic with equality, we have to add an extra constant Id, representing equality, to the operations. RRA denotes the class of algebras embeddable into direct products of algebras of the form

⟨𝔓(U × U), ∘,^{−1}, Id⟩

where Id = Id ↾ U = {⟨u, u⟩ : u ∈ U} is a constant of the expanded algebra.^{23} I.e.

RRA = SP {⟨𝔓(U × U), ∘,^{−1}, Id⟩ : U is a set} .

RRA abbreviates \underline{r}epresentable \underline{r}elation \underline{a}lgebras. RRA’s have been investigated more thoroughly than BRA’s; actually, Theorems 1,2 above were proved first for RRA’s.^{24} The proof of Theorem 1 applies almost verbatim to prove that RRA is a variety.

We note that BRA = SRdRRA, where Rd is the operator of taking appropriate reducts (cf. Remark 2). We also note that RRA is finitely axiomatizable over BRA, which means that by adding finitely many equations to the equational theory of BRA, we can obtain an axiomatization of RRA.^{25} Moreover, RRA is finitely axiomatizable over its ^{−1} and Id-free subreduct BSR, where a subreduct of K is a class of the form SRdK, cf. the convention for using SRd above item (3.3) (in Remark 2). This is in sharp contrast with the cylindric algebraic situation where each operation is only infinitely axiomatizable over the rest.^{26}. This also shows, that RRA being nonfinitely axiomatizable implies that neither BRA nor BSR is finitely axiomatizable.

20Notice that a BSR need not be closed under converse.

21Perhaps here [N85a], [N86] can be useful, because an analogous task was carried through there. The last 12 lines of Jónsson [J82, p. 276] seem to be also useful here.

22These BRA^{0}-like approximations of BRA-like “non-finitizable” classes are quite characteristic of algebraic logic, and they are relevant e.g. to algebraizations of syntactic aspects of logic like proof theory. For more on this see the Appendix (§8) and the subsection “The axiomatic approach… ” in the middle of §4.

23In Pratt [Pr90], the class RBM of representable Boolean monoids is obtained from our BSR’s by adding Id as an extra distinguished constant. So BSR’s are the Id-free subreducts of RBM’s. All the results mentioned above for BSR’s carry over to RBM’s; e.g. RBM is a discriminator variety, hence the simple RBM’s form a universally axiomatizable class, Theorems 1, 2 above apply to RBM etc. Some of the proofs (both for RBM and BSR) are analogous to that of Theorem 3 below. (These properties of RBM do not seem to be mentioned in [Pr90]).

24For historical notes on Theorem 1 for RRA’s see Thm.8.3(v) on p. 240 of [TG87].

25For issues concerning finite axiomatizability of a class over another one cf. [HMTII, p. 273 (Problems 5.4, 5.8)].

As a contrast we note that, letting _{∞}RRA denote the class of the infinitely representable RRA’s (i.e. where in ⟨𝔓(U × U) …⟩, |U| ≥ ω is required), Andréka proved that _{∞}RRA is not axiomatizable over RRA by any set of universal formulas containing finitely many variables, solving Problem 8 from [Gi89]. The notation _{∞}RRA is taken from [HMTII, p. 6].

In connection with algebraization of logic, we note that the so called Leibniz law of equality which says that equals cannot be distinguished, translates into the algebraic condition that Id is the neutral element of the semigroup operation “∘”.

Some special classes of RRA’s (RRA’s with pairing function elements, algebras in which below every relation there is a nonempty one which is a function, pair-dense RRA’s) turn out to be finitely axiomatizable, cf. Maddux [Ma78a, Ma87], Givant [Gi88].^{27} The elegant, purely algebraic proofs in these papers are examples for significant applications of algebra to logic, via connections between algebra and logic indicated in [TG87], in the Thm.13 through Prop. 16/a in §II.1 of [ANS84], and [HMTII, 4.3.29, p. 161]. Another fruitful direction for obtaining finite axiomatizability results for algebras of relations was initiated in cylindric algebra theory by Diane Resek in 1969; the binary relation oriented counterpart of which are the finite axiomatizability results for “relativized” RRA’s in e.g. Marx [Ma93], [Ma82], Kramer [K89].^{28} Further positive results, e.g. decidability for relativized RRA’s are in [N87a], [S89]. For Resek’s fundamental result a useful reference is Andréka– Thompson [AT88]; but see also [Ma89d], Monk [M91, Thm.9.4], Resek–Thompson [RT91], [HMTII, pp. 101, vi, 300].

The relevance of RRA’s to predicate logic has been elaborated in the book [TG87], but see also Maddux [Ma83] and §5.3 in conjunction with §4.3 of [HMTII].

RRA’s are also relevant to what is known as category theoretical logic. Namely, every RRA 𝔄 ⊆ ⟨𝔓(U × U), …, Id⟩ is equivalent with a category C whose objects are the subsets of U and whose morphisms are the binary relations between these sets. This C is actually more than a “plain” category, it is an enriched category, but it is exactly the enriched categories which play a central rôle in category theoretical logic, cf. e.g. Daigneault [D69], Zlatos [Z83, Z84], Freyd–Scedrov [FS90]. The objects of C can be recovered from 𝔄 as those elements of 𝔄 which are below the constant Id. The morphisms of C are the elements of 𝔄 in general, the domain of R is computed by the term Id ∧ (R ∘ 1) where 1 is the greatest element (in our case U × U) of any BA or BAO. Further, any homomorphism between two RRA’s 𝔄 and 𝔄′ gives rise to a functor between the corresponding enriched categories C and C′ (and vice versa).

DECIDABILITY, RELATIVIZED VERSIONS, NON–SQUARE APPROACH.

A class K of algebras is called decidable iff its equational theory Eq(K) =^{def} {e : K ⊧ e, e is an equation} is decidable.

In algebraic logic, the question of decidability of equational theories is an important one. Besides asking whether, say, RRA (or BA or whichever algebras of relations we are contemplating) is decidable, it is also important to ask which varieties contained in RRA or containing RRA are decidable. Recent works in this direction include [KNSS], [KNSS2], [SSNK], Marx [Ma93], [AGiN92], Jipsen [Ji92], [N96]. Cf. also Thm’s 6,7 herein.

We will see in Exercises 2 below that most of the classes studied so far are undecidable. E.g. the variety V_{+} defined in Item (3.1) above Theorem 1 is already undecidable, moreover, it remains such if we add equations valid in RRA, see [KNSS2] for the proof. On the other hand, if we omit the axiom of “∘” distributing over join, then the class becomes decidable (a result of V. Gyuris), cf. [KNSS2]. The variety of BAO’s (of any fixed similarity type) is also decidable and has many decidable subvarieties, cf. e.g. [Ji92].

26See Andréka’s negative solution for Problem 5.4 of [HMTII] to be mentioned later in this paper 27The possibility of extending Maddux’s approach to algebras of relations of higher ranks was investigated in [AN88].

Further, Givant generalized the positive results concerning pair dense algebras to a broader class he calls locally small RA’s, cf. [Gi88].

28For the definition (and brief discussion) of what we call relativized versions of the various kinds of algebras of relations, see the (first half of the) “Advanced topics in RCA theory” subsection of §4 herein. These algebras became extremely important in recent applications, cf. e.g. Masuch and Polos [MP92] or van Benthem [vB92].

Some of the reducts of RRA are decidable, while others are not. E.g. omitting complementation from RRA makes it decidable, cf. [A90a]. Further, let c_{0}(x) = 1∘x. Then e.g. the reduct containing the Booleans, ^{−1}, Id, and c_{0} only is decidable (a version of this is proved in [HMTII], a simpler proof is available from Andréka). Interestingly, most of what are called relativized versions of algebras of relations turn out to be decidable (even if we add new operators not available in RRA, cf. e.g. Marx [Ma93], [KNSS2], [N96], [N96a], Thm’s 6,7 herein). (We guess, the class of algebras in Thm.8/1 will also turn out to be decidable.) These “relativized versions” are based on the following idea.

When looking for a natural concept of an algebra of relations, we need not necessarily require the greatest element of an algebra of relations to be a Cartesian square (i.e. of form U × U). Of course, this greatest element has to be a relation (since all elements are supposed to be relations), but perhaps not necessarily of the form U × U. This approach is called the “non–square approach” by Yde Venema (cf. Venema [V91]), which might be a more suggestive name than the traditional “relativized version”. The non-square approach still yields algebras consisting of real relations, so it is a reasonable and healthy part of the quest for natural algebras of relations outlined in §2. A surprising result in Marx [Ma93a] says that all Boolean Algebras with Operators (BAO’s) are isomorphic to such non-square algebras of relations (if there are only finitely many operations in our algebras).^{29} This seems to point in the directon that the unifying power and circle of applicability of algebras of relations is considerably greater than we thought (which in turn was already much greater than what one would naturally expect). For more on the non-square (or relativized) approach cf. in §4 herein (subsection “Advanced topics in...”).

In a later version of this work we plan to treat decidability and also the non-square approach more systematically than in the present one.

EXERCISES 2. (2.1) Change Exercise 2 in the middle of §3 the following way. Let c : P(U) → P(U) be defined

by

c(x) = \{ \begin{matrix}U, if |x| > 1, \\ x, else, \end{matrix}

for any x ⊆ U. Prove that 𝔄 = ⟨𝔓(3), c⟩ is not contained in any discriminator variety, where 3 = {0, 1, 2}. Hint: Let 𝔅 ⊆ 𝔄 have universe {∅, 3, {2}, {0, 1}}. Then c{2} = {2} while c{0, 1} = 1^{𝔄}. Conclude that 𝔅 is subdirectly irreducible but not simple. Complete the proof as in Exercise 6.

(2.2) Let c be as in (2.1). Prove that K = S{⟨𝔓(U), c⟩ : U is a set} is not a discriminator class,

though K ⊆ SPSir(K). Hint: Use experience gained in (2.1). (2.3) Prove that BSR is a discriminator class. (2.4) Using (2.3), imitate the proof of Thm.1 for proving that BSR is a discriminator variety. (2.5) Consider Pratt’s class RBM defined in a footnote at the definition of RRA above. Execute

(2.3), (2.4) for RBM and for RRA. In particular, prove that RBM is a discriminator variety. (2.6) If too hard, this one may be skipped. But try first. Let K_{0} = SP{⟨𝔓(U × U),^{−1}⟩ :

U is a set}.

– Prove that K_{0} is a discriminator class.

– Imitating the proof of Thm.1, prove that K_{0} is a discriminator variety.

– Prove that K_{0} is finitely axiomatizable. Write up the axioms.

29This representation becomes especially natural if any two extra-Boolean operators are so called conjugates of each other (this does not restrict the ranks of these operators).

– Prove that the equational theory Eq(K_{0}) is decidable. Hint: First find out what SirK_{0} is.

(2.7) By a quasi-equation q(\overline{x}) we mean an implication of equations like [e_{1}(\overline{x}) ∧ e_{2}(\overline{x}) ∧ ⋯ ∧

e_{n}(\overline{x})] → e_{0}(\overline{x}).

– Prove that to every quasi-equation q(\overline{x}) of BRA’s there is an equation e(\overline{x}) such that Sir(BRA) ⊧ (q(\overline{x}) ↔ e(\overline{x})).

– Consider e associated to q above. Let “e” be denoted by f(q). Choose a function f which to each quasi-equation q associates an equivalent equation f(q). Prove that f can be chosen to be computable.

– Prove for the above defined computable function that (BRA ⊧ q) ⟺ (BRA ⊧ f(q)) for each quasi-equation q.

– Prove that for any class K = HK ⊆ BRA, (K ⊧ q) ⟺ (K ⊧ f(q)), for each quasiequation q.

– Prove from the above that for any subvariety K of BRA, the equational theory Eq(K) of K is decidable iff the quasi-equational theory Qeq(K) is decidable. (Do same for recursive enumerability instead of decidability.)

(2.8) Prove that Eq(BRA) is undecidable. Hint: Let Sgr be the class of semigroups. The classical result that Qeq(Sgr) is undecidable can be found in almost any textbook on logic. (Don’t prove this. Use it only.) Use this to prove that Qeq(BRA) is undecidable the following way. Observe that Sgr = SRdBRA if the only operation Rd does not forget is “∘”. Conclude that for a semigroup theoretic quasi-equation q, we have Sgr ⊧ q ⟺ BRA ⊧ q. Now, use (2.7) to conclude undecidability of Eq(BRA).

(2.9) Prove that BSR, RBM, RRA are all undecidable. Hint: check that the proofs for (2.7), (2.8)

go through unchanged for any of these classes. (2.10) Generalize (2.7) to arbitrary discriminator varieties. (2.11) If too hard, this should be skipped. Do not waste much time on this at first reading.

– Prove that the class V_{+} defined by condition (3.1) is not a discriminator class.

– Prove that Eq(V_{+}) is undecidable all the same. Hint: Use the methods in Gyuris-Kurucz-Németi-Sain [GyKNS].

– Prove that this remains so even if we remove the last sentence from (3.1), i.e. if we do not require that ^{−1} is a BA homomorphism or that x ∘ 0 = 0 ∘ x = 0. Hint: Same as above.

§4. Algebras for logics with equality

By this point we might have developed some vague picture of how algebras of binary relations are introduced, investigated etc. One might even sense that they give rise to a smooth, elegant and very exciting, powerful theory. However, our original intention was to develop algebras of relations in general, which should surely incorporate not only binary but also ternary, and in general n–ary relations.

Let us see how to generalize our RRA’s and BRA’s to relations of higher ranks. As we said, we would like the new algebras to be expansions of RRA’s (and BRA’s), or something like this. However, defining composition of n–ary relations for n > 2 is complicated^{30} . Therefore the following sounds like a more attractive idea: We single out the simplest basic operations on n–ary relations, and hope that composition will be derivable as a term–function from these. Let us see how we could generalize our generic or full RRA’s ⟨𝔓(U × U), ∘,^{−1}, Id⟩ to relations of rank n, for n ∈ ω. The obvious part is that these algebras will begin with ⟨𝔓(U × U × ⋯ × U), Id, …⟩, where Id = {⟨u, u, …, u⟩ : u ∈ U} is the n–ary identity relation. Again, Id is a constant, just as it was in the RRA case. Let ^{n}U denote U × U × ⋯ × U, e.g. ^{3}U = U × U × U. The new operations (besides the Boolean ones and Id) we will need are the algebraic counterparts of quantification ∃v_{i}, for i < n. So, we want an operation that sends the relation defined by R(v_{0}, v_{1}) to the one defined by ∃v_{0}R(v_{0}, v_{1}), and similarly for ∃v_{1}. For R ⊆ U × U let Dom(R) and Rng(R) denote the usual domain and range of R. For n = 2 we define c_{0}(R) = U × Rng(R) and c_{1}(R) = Dom(R) × U. Now ⟨𝔓(U × U), c_{0}, c_{1}, Id⟩ is the full cylindric set algebra of binary relations over U, for short the full Cs_{2}.

30Composition for n-ary relations has been studied in Marx–Németi–Sain [MNS96], Marx [Ma94]. The definition is ∘(R_{1}, …, R_{n}) = {⟨a_{1}, …, a_{n}⟩ : ∃x(⟨a_{1}, …, a_{n−1}, x⟩ ∈ R_{1} & … & ⟨x, a_{2}, …, a_{n}⟩ ∈ R_{n}}.

FIGURE 1

Before turning seriously to n–ary relations, we need the following:

Convention. Throughout we will pretend that Cartesian products and Cartesian powers are associative such that e.g.: ^{2}U ×^{3}U = ^{5}U, ^{n}U ×^{m}U = ^{n+m}U, and if R ⊆ ^{3}U then 2U × R ⊆ ^{5}U ⊇ R ×^{2}U.

The full Cs_{n} i.e. the full cylindric set algebra of n–ary relations is the natural generalization of Cs_{2} as follows.

Let R ⊆ ^{n}U. If Rng(R) = {⟨b_{1} … b_{n−1}⟩ : ⟨b_{0}b_{1} … b_{n−1}⟩ ∈ R for some b_{0}} then c_{0}(R) = U × Rng(R) considered as a set of n-tuples. Similarly, let Dom(R) = {⟨b_{0} … b_{n−2}⟩ : ⟨b_{0} … b_{n−2}b_{n−1}⟩ ∈ R for some b_{n−1}}, and let c_{n−1}(R) = Dom(R) × U. Generalizing this to c_{i} with i < n arbitrary, we obtain

c_{i}(R) = {⟨b_{0}, …, b_{i−1}, a, b_{i+1}, …, b_{n−1}⟩ : ⟨b_{0}, …, b_{n−1}⟩ ∈ R and a ∈ U} .

c_{i} is one of the most natural operations on relations. It simply forgets the i-th argument of the relation, or in other words, deletes the i-th column. However, since deleting the i-th column would leave us with an (n − 1)–ary relation, Dom(R) if i = n − 1, we replace the i-th column with a dummy column i.e. in the i = n − 1 case we represent Dom(R) with the “pseudo n–ary relation” Dom(R) × U. The “real rank” of an R ⊆ ^{n}U is always easy to recover, namely it is Δ(R) = {i < n : c_{i}(R) ≠ R}. So c_{i} is the natural operation of removing i from the (real) rank of a relation.

For example, c_{father} when applied to the “father, mother, child” relation gives back the “mother, child” relation coded as “anybody, mother, child” (in which the anybody argument carries no information i.e. is dummy). If U = {a, b, c} then c_{1}{⟨a, b, c⟩} = {⟨a, a, c⟩, ⟨a, b, c⟩, ⟨a, c, c⟩} and Id = {⟨a, a, a⟩, ⟨b, b, b⟩, ⟨c, c, c⟩}. By a full Cs_{n} we understand an algebra

ℜ𝔢𝔩_{n}(U) =^{def} ⟨𝔓(^{n}U), c_{0}, …, c_{n−1}, Id⟩

for some set U. By a Cs_{n} we understand a subalgebra of a full Cs_{n}. By a representable cylindric algebra of n–ary relations (an RCA_{n}) we understand a subalgebra of a direct product of full Cs_{n}’s (up to isomorphism), formally:

RCA_{n} = SP{ℜ𝔢𝔩_{n}(U) : U is a set} = SP(full Cs_{n}) = SPCs_{n}.

By the same argument as in the case of BRA’s, every RCA_{n} is directly representable as an algebra of n–ary relations (with the greatest relation a disjoint union of Cartesian spaces). RCA_{n} is one of the “leading candidates” for being the natural algebra of n–ary relations. The abstract algebraic picture is simple, an RCA_{n} is a BA together with n closure operations (in the usual abstract algebraic sense defined on the Boolean ordering) and an extra constant. Recall that a \underline{closure operation} c on a partially ordered set ⟨P, ≤⟩ is an order preserving and idempotent function c : P ⟶ P such that c(x) ≥ x. Boolean orderings with closure operations on them are one of the central concepts of abstract algebra, for example topological spaces or subalgebras of an algebra are often represented as such (e.g. if 𝔄 is an algebra, consider ⟨P(A), c⟩, where c(X) is the subalgebra of 𝔄 generated by X for X ⊆ A). A natural question comes up: Can these simple RCA_{n}’s recapture the power of RRA’s? To answer this question, consider the \underline{d}ummy \underline{r}epresentation Dr : P(U × U) ↣ P(U × U × U) of binary relations with ternary ones discussed above, i.e. Dr sends R to R × U. Then it is not hard to define terms τ^{∘} and τ^{−1} in the language of RCA_{3}’s such that Dr(R ∘ S) = τ^{∘}(Dr(R), Dr(S)) and Dr(R^{−1}) = τ^{−1}(Dr(R)) (see [HMTII, §5.3])^{31}. So in a sense RRA’s form a kind of a reduct of RCA_{n}’s for n ≥ 3. Thus the answer is that RCA_{n}’s, n > 2, recapture the power of RRA’s. (On the other hand, RCA_{2}’s do not.) This was a natural requirement we expected to meet, namely that the theory of n–ary relations should be an extension of that of binary relations.

At this point we can state the counterparts of Theorems 1, 2.

31E.g. c_{2}[c_{1}(x ∧ c_{0}Id) ∧ c_{0}(y ∧ c_{1}Id)] is one possible choice for τ^{∘}(x, y).

THEOREM 3 ([HMTII, 3.1.103, p. 43]). RCA_{n} is a variety for all n.

Proof and related considerations: Our Theorems 1, 3 are proved in [HMTII] without relying too much on universal algebra. Therefore it is not mentioned there that these theorems are almost immediate consequences of a well known universal algebraic property of the so called discriminator varieties recalled in Appendix 2 herein from e.g. [BS81, §IV.9]. The simple proof of the following basic property of discriminator varieties (together with the relevant definitions) is available in Appendix 2 herein: For any class K = SK of algebras with a discriminator term, if K is closed under ultraproducts then SPK is a variety. Now, the choice K = Cs_{n} is easily seen to satisfy these conditions as follows. Clearly Cs_{n} = SCs_{n}. To see that Cs_{n} has a discriminator term, let τ(x, y) = c_{0} … c_{n−1}(x⊕y) where ⊕ is Boolean symmetric difference. Let t(x, y, z) = (x∧τ(x, y))∨(z−τ(x, y)), where (x − y) abbreviates (x ∧ −y). Then t is a discriminator term in Cs_{n}, see Appendix 2.^{32} Cs_{n} is closed under ultraproducts (up to isomorphisms of course) because it is a pseudo-elementary class, i.e. if we add the base sets U as an extra sort to our Cs_{n}’s obtaining two sorted structures like ⟨𝔄, U, c_{i}, Id, ∈⟩ where 𝔄 ⊆ 𝔓 (^{n}U) and ∈⊆ ^{n}U × A is an n + 1–ary relation between sorts U and A, then this enriched class becomes first–order axiomatizable, as it is easy to see. This proves Theorem 3.

THEOREM 4 ([HMTII, 4.1.3, p. 111]. The variety RCA_{n} is not finitely axiomatizable if n > 2.

An outline of the proof is elaborated (and its intuitive idea is explained) in Remark 5.2 way below. That proof is based on that of Theorem 5.1 which is the reason why the proof is postponed to come after Theorem 5.1.

Theorem 4 above, due to Monk [M69], was refined by Andréka, Comer, and Hodkinson [Ho96] (but see also [Bi89], [Ma89], [J69], [ST89]). Andréka proved the following three results for n ≥ 3. Any set Σ of universally quantified formulas axiomatizing RCA_{n} has the following complexity property. For any k ∈ ω, there are infinitely many equations e ∈ Σ such that all basic operations^{33} of RCA_{n} together with more than k variables occur in e. Cf. [A91, Thm.6].^{34} RCA_{n} is not finitely axiomatizable over (the full first–order theory of) its Id-free subreduct (i.e. over the variety generated by the ⟨𝔓(^{n}U), c_{0}, …, c_{n−1}⟩’s). Neither is it finitely axiomatizable over its RRA subreduct. To formalize the latter, let the terms τ^{∘} and τ^{−1} discussed above be chosen such that their effect is the following: for R, S ⊆ U×U×U, τ^{∘}(R, S) = [Dom(R)∘Dom(S)]×U and τ^{−1}(R) = (Dom(R)^{−1})×U (where recall that Dom : P(U × U × U) ⟶ P(U × U) ). Then RCA_{3} is not finitely axiomatizable over the universally quantified theory of the algebras of the form ⟨𝔓(U × U × U), τ^{∘}, τ^{−1}, c_{2}, Id⟩. The latter of course is the reduct obtained from RCA_{3}’s by adding the term functions τ^{∘} and τ^{−1} as new basic operations, and forgetting c_{0} and c_{1}. A similar result applies for n ≥ 3. As a by-product, Andréka obtained a rather simple proof for Theorem 4. Comer ([Co89a]) proved that no generalized subreduct K of RCA_{n} containing all the cylindric algebraic operations except complementation is finitely axiomatizable for n > 2.^{35} (The point in talking about generalized reducts is that derived operations of RCA_{n} like c_{i}^{∂} = −c_{i}− may occur in K.) The question is open for n = 2. Hodkinson [Ho96] proved that RCA_{n} cannot be axiomatized by so-called Sahlqvist formulas, and in particular, it cannot be axiomatized by a set of positive in the wider sense equations^{36} Convention. An algebra ℜ is defined to be a generalized reduct of 𝔄 iff their universes coincide (R=A), and the operations of ℜ are term functions of 𝔄. I.e. iff ℜ is of the form ⟨A, τ_{i}⟩_{i∈I}, where τ_{i} is a term function of 𝔄. A class K_{1} is a generalized reduct of K_{2} (formally K_{1} = RdK_{2}) iff there is a uniform choice of the term functions ⟨τ_{i} : i ∈ I⟩ along which each member of K_{1} is a generalized reduct of one of K_{2}, and all such reducts from K_{2} are in K_{1}. A generalized subreduct is defined to be a subalgebra of a generalized reduct. From now on we will drop the adjective “generalized”. So from now on, by a reduct we will understand a generalized reduct (unless otherwise specified explicitly). More importantly, by a subreduct we will understand a generalized subreduct. We will use the convention for SRd, SPRd as introduced in Remark 2, but with the understanding that Rd means the generalized reduct. In particular, RdK_{2} is ambiguous because we must rely on context to specify the language (or similarity type) of RdK_{2}. An example for such a context is K_{1} = RdK_{2}, as explained in Remark 2.

32See e.g. [BS81, Example 2 below Def. 9.3 in §IV.9] and [AJN88] for more on CA’s being discriminator algebras.

33I.e. enough operations for generating the Booleans, all the c_{i}’s, and also Id.

34This solves a problem on p. 342_{8−4} of Monk [M69] (there the problem was credited to W. Craig). In passing we note that Andréka also solved the problem preceding the quoted one in [M69] asking if SNr_{n}CA_{n+k} is finitely axiomatizable. The rest of the problems in [M69] are still open.

35As a contrast, Andréka proved that the equational theories of RCA_{n} and RRA without complementation are decidable. (Using the terminology of [HMTI, 2.7.14, p. 439] validity of positive equations in RCA_{n} or RRA is decidable.) Cf. Andréka [A90a]. In these results Andréka includes the constants 0 and 1 into the similarity types of our algebras.

36These are equations in which complementation may occur only in front of constant terms. E.g. −Id can occur, but −x cannot occur in a positive in the wider sense equation.

\underline{Examples:} ⟨𝔓(^{3}U), τ^{∘}, τ^{−1}, c_{2}, Id⟩ discussed above is a reduct of the full Cs_{3} ⟨𝔓(^{3}U), c_{0}, c_{1}, c_{2}, Id⟩ in the new sense, but it was not such in the old sense. Thus RRA’s are subreducts of RCA_{3}’s in the new sense (but were not such in the old one). Further, RCA_{2}’s became subreducts of RRA’s now, but they were not such before (hint: c_{1}(x) = x ∘ 1, where 1 = U × U). A third important example of generalized subreducts of RCA_{3}’s will be Pinter’s substitution-cylindric algebras obtained by introducing the term functions s_{1}^{0}(x) = c_{0}(x ∧ c_{2}(Id)) (and similarly for s_{j}^{i}(x) with i, j < 3, i ≠ j), and considering the reducts ⟨𝔓(^{3}U), c_{0}, c_{1}, c_{2}, s_{j}^{i} : i, j < 3 and i ≠ j⟩. These algebras will be suitable for algebraizing logic without equality (so we will meet them in §5).

The ambiguous (or “context sensitive”) nature of the notation SRd is illustrated by RCA_{2} = SRdRRA and RRA = SRdRCA_{3} (especially when compared with RCA_{2} = SRdRCA_{3} and RCA_{2} ≠ RRA).

Recursive sets of equations characterizing RCA_{n} are given and discussed on pp. 112–119 of [HMTII], the first of these originating with Monk [M69]. Venema [V90], [V89, Def.3.3.3 & Thm.3.3.5], and Hirsch–Hodkinson [HH94] contain new results in this line. The situation is analogous to the BRA case described below Theorem 2. Finding mathematically transparent, simple sets of equations defining RCA_{n} is one of the central open problems, cf. Problems 4.1, 4.16 of [HMTII]. We should point out the important result of Diane Resek that a slight generalization of RCA_{n} denoted by CA_{n} ∩ICrs_{n} in [HMTI, p. 101] is finitely axiomatizable; see Andréka–Thompson [AT88] for a simple proof. ([Ma89d] and Resek–Thompson [RT89] are also in this line.)

The following is an important digression.

The axiomatic approach, “abstract” (not necessarily representable) cylindric algebras (CA’s), relation algebras (RA’s) etc.

By Theorems 2, 4, the varieties RCA_{n}, BRA, and RRA, whose elements consist of concrete relations (up to isomorphism of course) are not finitely axiomatizable (non-finitizable, for short). Let us take the example of, say, RCA_{n}. Because of the above non-finitizability property of RCA_{n}, there is a conventional agreement in the literature for studying a fixed finitely axiomatizable variety CA_{n} approximating RCA_{n}. The members of CA_{n} are called “(n–ary) cylindric algebras”. By saying that CA_{n} approximates RCA_{n} we mean that CA_{n} ⊇ RCA_{n} and that CA_{n} is as close to RCA_{n} as we can get without losing attractiveness and mathematical transparency of our axioms describing CA_{n}. For example, the equations expressing that c_{i} is a closure operator are transparent enough, so we include them into the axiomatization of CA_{n}. Similarly for the equation c_{i}c_{j} = c_{j}c_{i} expressing that the c_{i}’s commute. On the other hand, consider the following statement (∗) about relations. For any R ⊆ ^{n}U, let Dom(R) = {s_{0} : s ∈ R}, i.e. Dom(R) is the smallest H such that R ⊆ H ×^{n−1}U. Now,

(∗) If R ⊆ ^{n}U and 0 < |Dom(R)| = m < n then it is impossible to have disjoint relations R_{0}, …, R_{m} ⊆ R such that (∀k ≤ m) c_{0}R_{k} = c_{0}R.

(∗) is translatable into a cylindric algebraic equation e_{(∗)} using m + 2 variables (standing for R_{0}, …, R_{m}, and R). This translation is based on the fact that |Dom(R)| = m can be expressed by using Id. This equation e_{(∗)} has been considered not elegant enough for being added as an axiom of CA_{n}’s.^{37}

If besides RCA_{n} we would like to see another example, then the property formulated in (3.1) is a transparent enough property of RRA such that it is postulated as an axiom of the finitary variety RA (relation algebras) approximating RRA. (A “not transparent enough” equation can be obtained from the above e_{(∗)} by choosing n = 6 and instead of expressing |Dom(R)| < 6 by Id which is impossible in RRA, letting ^{2}(Dom(R)) = S ∪^{.} T ∪^{.} Id with S, T symmetric, antireflexive relations such that T = (S ∘ S) ∖ Id and S = (T ∘ T) ∖ Id.)

The above circumscription of when we say that a finitizable variety V approximates a nonfinitizable one, say RV, is not a precise mathematical definition, but we hope it may have some heuristic value. The name of the approximated variety begins with an R (like RV, RCA_{n}, or RRA) to remind the reader that these are the members of V which can be \underline{r}epresented as algebras of \underline{r}eal \underline{r}elations (it is left to the reader to decide which of the three “r”-s should the R in RV stand for).

To each of the distinguished varieties like RRA, RCA_{n}, RPA_{n} etc. discussed in this paper, there corresponds in the literature a finitizable approximation RA, CA_{n}, PA_{n} etc. respectively, analogously to the approximation of RCA_{n} by CA_{n}. We will discuss these approximating varieties CA_{n}, RA etc. in the Appendix (§8), where we will also give references for their further study.

While the concrete algebras of relations like RCA_{n} or RRA provide the central tool for algebraizing the semantic aspects of logics, their finitary approximations like CA_{n} or RA are useful in algebraizing proof theoretic concepts, see [HMTII, §4.3] (and implicitly in [TG87]). From a different angle, the relationship between RCA_{n}, RRA etc. and CA_{n}, RA etc. seems to be somewhat analogous to that between the standard model 𝔑 = ⟨ω, +, ×, 0, 1⟩ of arithmetic and Peano’s Arithmetic. E.g. both the theory of Peano’s Arithmetic and CA_{n} (or RA) devote considerable effort to developing methods for constructing nonstandard models (i.e. non-representable algebras), cf. e.g. [HMTII, pp. 85–100]. Since much of the intuition behind the theories of Peano’s Arithmetic, CA_{n} and RA derives from the standard models (i.e. 𝔑, RCA_{n} and RRA), we will concentrate on the representable algebras RCA_{n}, RRA etc., and will deal with their finitary approximations CA_{n}, RA etc. only very briefly, in the present paper. For completeness, full definitions of CA_{n}, RA etc. as well as further information on these abstract (or “approximating”) classes are available in the Appendix (§8) of this paper.

∗ ∗ ∗ ∗ ∗

Let us return to the connections between RCA_{n} and \underline{algebraization of logic}.

Let 𝔐 be a model of a finite language (or signature). We may treat all symbols of the language as relation symbols as it is well known, cf. Bell–Slomson [BS69, §2.10 pp. 97–100]. So 𝔐 is of the form 𝔐 = ⟨M, R_{0}, …, R_{k}⟩. Let n be a strict upper bound of the ranks of R_{0}, …, R_{k}. We may safely treat each R_{i} as if it were of rank n by using our dummy representation Dr(R_{i}) = R_{i} × U × U if the rank of R_{i} is, say, n − 2. Now, R_{0}, …, R_{k} are elements of the full Cs_{n} ℜ𝔢𝔩_{n}(M) = ⟨𝔓(^{n}M), c_{0}, …, Id⟩. Let ℭ be the subalgebra of ℜ𝔢𝔩_{n}(M) generated by {R_{0}, …, R_{k}}. Now, ℭ is the algebraic counterpart (consisting of n–ary relations) of 𝔐 as a result of the standard algebraization of logic described in §4.3 of [HMTII] (where ℭ was denoted by ℭ𝔰_{n}^{(𝔐)}). One can check that ℭ consists of exactly those relations over M which are definable in 𝔐 with using at most n variables. It is natural that the algebraic counterparts of models (i.e. of the semantic part of first–order logic) should be algebras consisting of those relations that are definable in the original models. Indeed, it is these definable relations which provide meanings for the formulas of first–order logic; hence by using them we can arrive at Lindenbaum–Tarski algebras of equivalence classes of formulas (which are the so called formula algebras of theories), i.e. algebraic counterparts of syntactic notions of logic. Moreover, along these lines we can arrive at algebraic counterparts of not only syntactical and semantical notions of logic, but also at an algebraic counterpart of the syntax–semantics duality in logic, including the notion of axiomatizable classes of models.^{38} Of course, one can repeat the same process of algebraization with the same key ideas (algebras consisting of general relations conceived of as meanings of formulas) for different quantifier logics, e.g. first–order modal logics, higher order logics, logics with new quantifiers like “for many v_{i} φ”, abstract model theoretic logics from Barwise–Feferman [BF85]. The details are worked out explicitly in §4.3 of [HMTII] for classical first–order logic in particular, and in §5.6 for logics in general. See also Blok–Pigozzi [BP89, Appendix C] and Németi [N89].

37Actually from time to time some hesitation comes up concerning the status of this particular equation e_{(∗)}, and perhaps sometime in the future some reinforced hyper-strong subvariety HCA_{n} of CA_{n}, satisfying e_{(∗)}, might be introduced, but there are more complicated versions of e_{(∗)} which, very probably, will not be added to HCA_{n} either.:

Let |Dom(R)| = m and assume n < m < ω. Then there are more complicated (than using Id) ways of equationally “forcing” |Dom(R)| = m by introducing new relations which “count” the size of Dom(R). The so obtained variant of e_{(∗)} is a typical example of a known validity of RCA_{n} which does not count as transparent enough to be added as an axiom of the approximating variety CA_{n}.

Let us have a concrete look at RCA_{n}. The logic naturally corresponding to RCA_{n} is first–order logic L_{n} restricted to the first n individual variables. Besides [HMTII], see also [TG87, §§3.10, 7.3], where L_{n} is denoted as L_{n}, and sometimes as L_{n}^{+}, P_{n}. To illustrate the correspondence with logic, we note that Theorem 4 above implies a strong incompleteness theorem for L_{n}. Roughly speaking, this logical corollary of Theorem 4 says that no inference system given by a finite set of schemata of axioms and Hilbert–style derivation rules^{39} can be complete and strongly sound for L_{n}. Similarly, an algebraic theorem in [ACN] implies that L_{n} does not enjoy the Beth definability property.

In passing, we note the following. The Beth property has variants called weak Beth property and very weak Beth property. Kearnes-Sain-Simon [KSS96] proved that for L_{3} the weak Beth property fails while the very weak one holds. The proof goes via algebraic equivalents of these properties established in Sain [S89] and in [ANSK]. We conjecture that their proof will go through for n > 3, too. It remains an interesting open problem whether L_{2} has the weak Beth property.

For more on the algebraic counterparts of logical properties like Beth’s definability and Craig’s interpolation see e.g. Maksimova [Ms77, Ms88], [HMTII, pp. 259–260], [HMT, pp. 356–357, p. 178], [ANS84], [Co84], [TR87], [W84], [O86], Goranko [G85], Marx [Ma94], Hoogland [H96], Madarász [M96, M97], [ANS94a], Palasinska-Pigozzi [PP96].

38In this connection we note that, roughly speaking, the algebraic counterparts of the basics of logical syntax (i.e.

of the sets of logical formulas with the logical connectives acting on them) are the free algebras, cf. e.g. [AS78].

Accordingly, the free algebras of RCA_{n}, RRA, and of their finitizable approximations CA_{n} and RA (discussed above and in §8) have already received some attention. See e.g. Pigozzi’s results concerning free algebras in [HMT, HMTII]; [N86], [N85a] for free cylindric algebras (and free RCA_{n}’s), and Andréka–Jónsson–Németi [AJN88] for free relation algebras, free RRA’s, and free algebras in other forms of algebraic logic (but see also Blok [Bl76], Urquhart [U73] for the latter). The field of free algebras is difficult to develop, and therefore this area is far from being closed, there are many open problems, and unexplored territories here.

39Here we use these words in the traditional sense, which was formalized e.g. on p. 5 of Blok–Pigozzi [BP89].

Inference systems of this kind are sometimes called strongly sound Hilbert-style ones. A not strongly sound but complete inference system for L_{n} was obtained in Venema [V90, V91]. This is given by a finite schema (but the notion of an inference rule is different from ours or from that of [BP89]). Other, similarly not strongly sound but complete interence systems are in Mikulás [Mi95]. Venema’s and Mikulás’ results are “nonstandard” possible solutions for Problem 2.0 herein. For completeness we note the following. Let ⊢_{n} denote provability by one of the standard Hilbert style axiomatizations of L_{n} (cf. e.g. [HMTII] or [TG87]). Basically, ⊢_{n} is obtained by restricting the usual axiomatization of L_{ωω} to L_{n}. If we take the syntactic version of the Lindenbaum–Tarski algebra of L_{n} induced by ⊢_{n} then we obtain the free CA_{n} discussed in the preceding subsection (“The axiomatic approach… ”)

and in §8. If we take all ⊢_{n}-consistent theories of L_{n} and their Lindenbaum–Tarski algebras (modulo ⊢_{n} and not semantically) then we get exactly the members of CA_{n}.

PROBLEM 2.0. Find a nice complete inference system for L_{n}. In view of Theorem 4 above (and of its improvements by Andréka), the “nicest” kinds of inference systems will not work. As in [HMTII, Prob.4.16], “nice” implies that L_{n}-proofs of validities involve only valid formulas of L_{n}. (It would be also nice to have only a finite schema of rules and axioms. This problem, probably, will not have a unique final solution, but instead several possible solutions interesting from different points of view.) This problem is (at least implicitly) present in [HMTII] and [TG].

How far did we get in obtaining algebras of relations in general (binary, ternary, …, n–ary, …)? RCA_{n} is a smooth and satisfactory algebraic theory of n–ary relations. So, can our theory handle all finitary relations? The answer is both yes and no. Namely, since n ∈ ω is arbitrary, in a sense, we can handle all finitary relations. But, we cannot have them all in the same algebra or in the same variety. For any finite family of relations, we can pick n such that they are all in RCA_{n}. But this does not extend to infinite families of relations. To alleviate this, we could try working in the system ⟨RCA_{n} : n ∈ ω⟩ of varieties instead of using just one of these. To use them all together, we need a strong coordination between them. This coordination is easily derivable from the embedding function Dr sending R to R × U for R ⊆ ^{n}U defined above. Let 𝔄 ⊆ ℜ𝔢𝔩_{n}(U) = ⟨𝔓(^{n}U) ⋯⟩ be a Cs_{n} and let 𝔅 be the Cs_{n+1} generated by the Dr image of 𝔄, i.e. 𝔅 ⊆ ℜ𝔢𝔩_{n+1}(U) = ⟨𝔓(^{n+1}U) ⋯⟩ is generated by {Dr(R) : R ∈ A}. The biggest 𝔄 yielding the same 𝔅 is called the n–ary neat-reduct of 𝔅, formally 𝔄 = Nr_{n}(𝔅). It is easy to extend Nr_{n} in a natural way to all elements of RCA_{n+1}. Intuitively, Nr_{n}(𝔅) is the algebra of n–ary relations “living in” the algebra 𝔅 of n+1–ary relations. It is not hard to see that Nr_{n} : RCA_{n+1} ⟶ RCA_{n} is a functor, in the category theoretical sense, for every n. Now, we can use the collection of varieties RCA_{n} for all n ∈ ω, synchronized via the functors ⟨Nr_{n} : n ∈ ω⟩, as a single mathematical entity containing all finitary relations.

Another possibility is to insist that we want all finitary relations over U represented as elements of a single algebra. In other words, this goal means that instead of a system of varieties we want to consider a single variety that in some sense incorporates all the original varieties taken together. Indeed, each RCA_{n} can be viewed as incorporating all the RCA_{k}’s for k ≤ n, since the latter can be recovered from RCA_{n} by using the functors Nr_{n−1}, Nr_{n−2} etc. So as n increases, RCA_{n} gets closer and closer to the variety we want. Indeed, we take the limit of this sequence. There are two ways of doing this, the naïve way we will follow here and the category theoretical way we only briefly mention. It is shown in the textbook Adamek–Herrlich–Strecker [AHS] that the system or “diagram”

RCA_{1} ⟵^{Nr_{1}} RCA_{2} ⟵^{Nr_{2}} ⋯ RCA_{n} ⟵^{Nr_{n}} RCA_{n+1} ⋯

is “convergent” in the category theoretic sense, i.e. that it has a limit L. Indeed, it is this class L of algebras that we will construct below in a naïve way that does not use category theoretic tools or concepts.

We first extend our Convention, stated at the beginning of the present section concerning associativity of Cartesian products and powers. In the sequel ^{ω}U is the set of ω-sequences over U. Furthermore, ^{n}U ×^{ω}U = ^{ω}U, and if R ⊆ ^{n}U then R ×^{ω}U ⊆ ^{ω}U, for n < ω. We will also have to distinguish the constant Id of RCA_{3} from that of RCA_{4}. Therefore we let Id_{n} =^{def} {⟨a, …, a⟩ : a ∈ U} denote the n–ary identity relation on U.

How do we obtain an algebra containing all finitary relations over U? If R is binary, but we want to treat it together with a 5–ary relation then we represent R by R × U × U × U = R ×^{3}U in a Cs_{5}. Similarly, if we want to have R together with (n + 2)–ary relations then we represent R with R ×^{n}U in a Cs_{n+2}. Taking this procedure to the limit, if we want to treat R together with relations of arbitrary high ranks, then we can represent R with R ×^{ω}U. This way we can embed all finitary relations into relations of rank ω, and relations of different ranks become “comparable” and “compatible”. In particular we avoid the problem we ran into at the end of §2 in connection with the Boolean algebra 𝔓(^{<ω}U), because e.g. the complement of R × ^{ω}U is \overline{R} × ^{ω}U where \overline{R} denotes ^{n}U ∖ R if R is an n-ary relation. Instead of trying to tame the Boolean-like algebra {R : R ⊆ ^{n}U for some n} ⊆ P(^{<ω}U), we simply represent R ⊆ ^{n}U by R × ^{ω}U which is an element of the BAO (\underline{B}oolean \underline{a}lgebra with \underline{o}perators) ⟨𝔓(^{ω}U), c_{0}, …, c_{n}, …⟩_{n<ω}, where 𝔓(^{ω}U) is a BA, and c_{i} is defined exactly the same way as in the Cs_{n} case. We still haven’t obtained the definition of Cs_{ω}’s from that of Cs_{n}’s because we do not know what to do with the constant Id. More specifically, we want to be able to use the neat reduct functor Nr_{n}, as the inverse of R ↦ R ×^{ω}U for R ⊆ ^{n}U, in order to recover the original Cs_{n}’s from the new Cs_{ω}. This means that for Id_{n} ⊆ ^{n}U we want Id_{n} ×^{ω}U to be a derived constant (distinguished element) in our algebra. Adding Id_{ω} = {⟨a, …, a, …⟩ : a ∈ U} as an extra constant does not ensure this any more. One of the most natural solutions is letting Id_{ij} = {q ∈ ^{ω}U : q_{i} = q_{j}} and defining a full Cs_{ω} as

ℜ𝔢𝔩_{ω}(U) =^{def} ⟨𝔓(^{ω}U), c_{i}, Id_{ij}⟩_{i,j<ω} ,

where the Id_{ij}’s are constants. The price we had to pay for replacing the finite bound n on the ranks of relations we can treat with the infinite bound ω is that we had to break up our single constant Id to infinitely many constants Id_{ij} (i, j ∈ ω).

RCA_{ω} is defined to consist of all subalgebras of direct products of full Cs_{ω}’s (up to isomorphisms). Again, as it was the case with BRA’s and RCA_{n}’s, RCA_{ω}’s are directly representable as algebras whose elements are ω–ary relations.

THEOREM 5 ([HMTII, 3.1.103, p. 43]). RCA_{ω} is a variety.

On the proof: The proof is harder than those of Theorems 1, 3. The reason for this might be the fact that RCA_{ω} is not a discriminator variety unlike RCA_{n}, RRA, and BRA, which are discriminator varieties. In this connection we note that despite of this, many of the results of the theory of discriminator varieties carry over to RCA_{ω} because of the following. RCA_{ω} is “reductlocally discriminator” in the sense that for any finite part t_{0} of the similarity type t_{1} of RCA_{ω}, there is a t_{0} ≤ t ≤ t_{1} such that SRd_{t}RCA_{ω} is a discriminator variety.

Let 𝔅 ∈ RCA_{ω} and 𝔄 = 𝔅/Θ for some congruence Θ of 𝔅. For n ∈ ω, let Rd_{n}𝔄 be the reduct of 𝔄 to the language of RCA_{n}’s. It is not hard to see that Rd_{n}𝔅 ∈ RCA_{n}. By Theorem 3 then Rd_{n}𝔄 = (Rd_{n}𝔅) /Θ ∈ RCA_{n}. To prove Theorem 5, it is enough to prove that

(∗)  (∀n ∈ ω)Rd_{n}𝔄 ∈ RCA_{n} implies 𝔄 ∈ RCA_{ω}

for any 𝔄 ∈ HSPRCA_{ω}.

It is not very easy to prove (∗), but it is not hopelessly hard either. E.g. one can use ultraproducts in “pasting together” the representations of the (Rd_{n}𝔄)’s in order to represent 𝔄. For this the basic lemma [HMTII, 3.1.92, p. 36] discussing ultraproducts of Cs_{ω}’s and their generalizations Gs_{ω}’s is sufficient.

For completeness, we note that [HMTII, 3.1.103, p. 43] gives a direct proof for Theorem 5 not using discriminator varieties or other special tools of universal algebra. Further, it is useful to keep in mind that though RCA_{ω} is not a discriminator variety, it is an arithmetical (congruence distributive and congruence permutable) one. For the rather strong theory of the latter see [MMT, p. 247], [BS81]. Actually, every variety of BAO’s is arithmetical by [MMT, Thm.4.143] or [BS81, Thm.II.12.5] due to Pixley.

We note that RCA_{ω} is the variety generated by the class L which was the limit of the RCA_{n}’s above, see [AHS]. Can we actually recover the algebras of finitary relations from the huge full Cs_{ω}’s? Let Rf(U) = {R ×^{ω}U : R ⊆ ^{n}U for some n}. Then Rf(U) ⊆ P(^{ω}U); moreover it is a subalgebra of the full Cs_{ω} ℜ𝔢𝔩_{ω}(U) with universe P(^{ω}U); see [HMTII, Def.3.1.130, p. 55], where this subalgebra was denoted by ℜf(U, ω). We will use the notation ℜ𝔣(U). (The letters Rf (and ℜ𝔣) refer to “\underline{f}initary \underline{r}elations”.) Now, for the above mentioned class L we have L = S\{ℜ𝔣(U) : U is a set\} .^{40} In a sense, L is the narrowest reasonable class of algebras of finitary relations. The class L and its relationship with RCA_{ω} was systematically investigated in Andréka [A73], Andréka–Gergely– Németi [AGN73], [AGN77], [HMTAN], [HMTII]. In the first three works the class was denoted by Lv or Lr, while in the last two by Cs_{ω}^{reg} ∩ Lf_{ω}, the latter being the standard notation today. Cf. e.g. Shelah [Sh89], Monk [M89], Serény [Se86], for recent results not in [HMTII]. RCA_{ω} = HSPL is Thm.3.1.123 of [HMTII]. [HMTAN] proves that the smallest quasivariety containing L is also RCA_{ω} but RCA_{ω} ≠ SPL, i.e. the infinitary quasivariety generated by L is strictly smaller than RCA_{ω}.

40For this equality to be literally true, when forming the category theoretic limit L, instead of the varieties RCA_{n} we have to start out from their subdirectly irreducible members, which are nothing but Cs_{n}’s. So L is the limit of the sequence Cs_{1}, …, Cs_{n}, . . ..

We return briefly to \underline{algebraization of logic}: RCA_{ω} is the true algebraic counterpart of full first– order logic (with equality). Within RCA_{ω}, the algebraic counterparts of models of first–order logic are the members of L. Indeed, let 𝔐 = ⟨M, R_{i}⟩_{i∈I} be a first–order model. Then the subalgebra of ℜ𝔣(M) generated by {R_{i} × ^{ω}M : i ∈ I} is the algebraic counterpart of 𝔐.^{41} This is not very surprising after having seen that the logical counterpart of RCA_{n} was first–order logic L_{n} restricted to the first n variables. It is natural to expect that removing the bound n on the ranks of the relations will result in removing the bound on the number of variables, in the corresponding first–order logic.

The algebraic counterparts of the logical connectives are operations of RCA_{ω}. (This is standard in algebraic logic; see [HMTII, §5.6], or Blok–Pigozzi [BP89], Andréka–Sain [AS78]), or Andréka– Blok–Németi–Pigozzi–Sain [ABNPS]. The algebraic counterpart of ∃v_{i} is c_{i}, and that of v_{i} = v_{j} is Id_{ij}. With these, substitution of variables is expressible; namely the result of substituting v_{j} for all free occurrences of v_{i} in the formula φ is equivalent to ∃v_{i}(v_{i} = v_{j} ∧ φ); cf. [HMTII, §4.3]. In particular, R(v_{1}v_{1}) is ∃v_{0}(v_{0} = v_{1} ∧ R(v_{0}v_{1})), R(v_{1}v_{0}) is ∃v_{2}[v_{2} = v_{0} ∧ ∃v_{0}(v_{0} = v_{1} ∧ ∃v_{1}[v_{1} = v_{2} ∧ R(v_{0}v_{1})])]. Therefore we do not need to introduce an algebraic operation to represent substitutions of individual variables as we did in the case of BRA’s, by means of ^{−1}.

So, we know what the algebraic counterparts of logical connectives and of individual models are. The algebraic counterparts of classes of models (especially of axiomatizable classes) are certain elements^{42} of SPL, cf. [HMTII, p. 168] and Németi [N78]. The algebraic counterpart of a class K of models was denoted by 𝔊𝔰^{K} in [HMTII].

Moving in the opposite direction, the logical counterparts of cylindric equations (i.e. equations in the language of RCA_{ω}’s) are not concrete first–order formulas, as one would expect, but instead they are schemata of formulas, see Németi [N87]^{43}. A typical example for a schema is φ → ∃v_{1}φ, where φ is a variable ranging over formulas. The corresponding cylindric equation is x ≤ c_{1}x. Examples for logical investigations where schemata of formulas play a central rôle are the recent Bernarducci [B89], [B89a], Vardanyan [V86], Smorynski [Sm85], [Sm84].

In more detail, the connection between first–order logic (also some of its generalizations) and RCA_{ω} theory (including its generalizations) is elaborated in [HMTII, §4.3], Németi [N89], [N78], Blok–Pigozzi [BP89, Appendix C], [ABNPS]. (M. Rubin has results concerning decidability etc. of equational theories of RCA_{ω}’s associated to various theories of logic.)

An interesting and useful logical counterpart different from the ones emphasized in the present paper for RCA_{n} and RCA_{ω} was introduced and studied in Venema [V89, V90]. This logical equivalent of RCA is a propositional multi-modal logic. This approach looks rather promising.

41For certain purposes the algebraic counterparts of models are not elements of the class L, but rather homomorphisms from free algebras (namely, free RCA_{ω}’s over various generator sets) into elements of L; cf. [HMTII, pp. 256, 257_{9}]; [AS78, pp. 46–48], where connections with Initial Algebra Semantics of Computer Science are explained; and Halmos [Ha85, p. 208 lines 6–7].

42This claim might puzzle some readers because classes of models are “big” when compared to single models (so how come they both translate to single algebras). In this connection we would like to point out that in a certain algebraic sense the elements of SPL are big when compared to those of L. (The elements of L are subdirectly irreducible while those of SPL are not.)

43At least this is the case in many situations. A case when the logical counterparts can be treated as concrete formulas is [HMTII, Thm.4.3.57].

The theory of RCA_{ω} is extensively developed (cf. [HMT, Chapters 1–4], and [HMTAN]). The few results we can mention here are very far from forming even a representative sample. The lattice of subvarieties of RCA_{ω} together with their decision problems is investigated in [N87], [N85]. Open problems (about subvarieties etc.) that might be rewarding to work on are in [HMTAN], and [HMTII, pp. 179–180]. E.g. Problem 4.2 in the latter asks if there are 2^{|α|} many subvarieties of RCA_{α} for |α| > ω, where RCA_{α} is the natural generalization of RCA_{ω} to be defined soon. (A list indicating the status of the problems raised in [HMT], [HMTII], [HMTAN] is available from Monk, and has been published in [AMN].) Sain [S89] contains, besides recent results, important and natural open questions. One of them asks for a universal algebraic (or category theoretic) characterization of the full Cs_{ω}’s as members of the variety RCA_{ω}. The analogous problem is also open for RCA_{n} and RRA. An important recent direction is taken in Monk [M89]. There is a Galois theory of RCA’s, see e.g. Comer [Co84], Daigneault [D64], Driessel [Dr68], Reyes [R70], [M89], Plotkin [Pl88, Pl89]. We should also mention the works of Serény [Se85], [Se86], Ferenczi [F89, F89a], Shelah [Sh89], Biró–Shelah [BiS88].

Theorems 2 and 4 above,which say that BRA, RRA, and RCA_{n} are not finitely axiomatizable, carry over to RCA_{ω} too, but to avoid triviality, instead of non-finite axiomatizability we have to state something stronger, because RCA_{ω} has infinitely many operations and finitely many axioms can speak about only finitely many operations anyway. Taking this into account, when trying to axiomatize RCA_{ω}, one could still hope for a finite “schema” (in some sense) of equations treating the infinity of the RCA_{ω}-operations uniformly. A possible example for a finite schema is (E0–7) in the proof of Proposition 8.3 in §8 (Appendix). The following theorem, generalizing a very important result of Monk ([HMTII, 4.1.7]), implies that it will be hard to find such a schema, and that certain kinds of schemata are ruled out to begin with.

THEOREM 5.1 (Andréka). The variety RCA_{ω} is not axiomatizable by any set Σ of universally quantified formulas such that Σ contains only finitely many variables.^{44}

Proof:

Plan: For all n ∈ ω we will construct an algebra 𝔄_{n} such that

a) 𝔄_{n} ∉ RCA_{ω}

b) every n–generated subalgebra of 𝔄_{n} is in RCA_{ω}.

This will prove the theorem because of the following. Assume that Σ is a set of quantifier-free formulas such that Σ contains at most n variables (|var(Σ)| ≤ n < ω) and RCA_{ω} ⊧ Σ. Then 𝔄_{n} ⊧ Σ is easily seen as follows. Every n-generated subalgebra of 𝔄_{n} is in RCA_{ω}, and the validity of Σ in any algebra 𝔅 depends only on the validity of Σ in the n-generated subalgebras of 𝔅, because |var(Σ)| ≤ n and Σ contains no quantifiers. However, 𝔄_{n} ∉ RCA_{ω}, thus Mod(Σ) ⊈ RCA_{ω} showing that Σ does not axiomatize RCA_{ω}.

Just as we write −x instead of −(x), we will often write c_{i}x for c_{i}(x).

Construction of 𝔄_{n}: Let m ≥ 2^{n} be finite, and let ⟨U_{i} : i < ω⟩ be a system of pairwise disjoint sets each of cardinality m. (I.e. (∀i < j < ω)[|U_{i}| = m and U_{i} ∩ U_{j} = ∅].) Let

U = ⋃{U_{i} : i ∈ ω}, let

q ∈ ×_{i∈ω}U_{i} =^{def} {s ∈ ^{ω}U : (∀i ∈ ω)s_{i} ∈ U_{i}} be arbitrary,

R = {z ∈ ×_{i∈ω}U_{i} : |{i ∈ ω : z_{i} ≠ q_{i}}| < ω}, and let

𝔄′ be the subalgebra of ⟨𝔓(^{ω}U), c_{i}, Id_{ij}⟩_{i,j∈ω} generated by the element R.

44She also proved the following stronger result. Let Σ be a set of universally quantified formulas axiomatizing RCA_{ω}, and k ∈ ω. Then there is Σ_{k} ⊆ Σ such that each e ∈ Σ_{k} contains more than k c_{i}’s, more than k variables, all the Booleans, and at least one Id_{ij}. Cf. [A91, Thm.6].

Then R is an atom of 𝔄′ because of the following. For any two sequences s, z ∈ R there is a permutation σ : U ↣ → U of U taking s to z and fixing R, i.e. s ∘ σ = z and R = {p ∘ σ : p ∈ R} (the obvious choice for σ, interchanging s_{i} and z_{i} for all i ∈ ω and leaving everything else fixed, works). If σ is a permutation of U fixing R, then σ fixes all the elements generated by R because the operations of RCA_{ω} are permutation invariant. Thus if ∅ ≠ a ∈ A′ and s ∈ a ∩ R then R ⊆ a, showing that R is an atom of 𝔄′.

We now “split R into m + 1 new atoms R_{j} each imitating R” obtaining a new, bigger algebra 𝔄 from our old 𝔄′. I.e. we choose a larger algebra 𝔄 containing 𝔄′ as a subalgebra (𝔄 ⊇ 𝔄′) such that 𝔄 satisfies the conditions below and is otherwise arbitrary.

𝔄′ ⊆ 𝔄, the Boolean reduct of 𝔄 is a Boolean algebra, R_{j} are atoms of 𝔄 and c_{i}R_{j} = c_{i}R for j ≤ m, i < ω, each element of 𝔄 is a join of an element of 𝔄′ and of some R_{j}’s c_{i} distributes over joins, for any i < ω, i.e. 𝔄 ⊧ c_{i}(x ∪ y) = c_{i}x ∪ c_{i}y.

Note that in 𝔄 “∪” is only an abstract algebraic operation and not necessarily set theoretical union. It is easy to see that such an extension 𝔄 of 𝔄′ exists.

FIGURE 2

By the above, we have constructed our algebra 𝔄_{n} =^{def} 𝔄. Convention: In this proof we use the symbols ∩, ∪ denoting the concrete operations of our set algebras (Cs_{ω}’s) also as the corresponding abstract algebraic operation symbols (denoting themselves in Cs_{ω}’s). So, if x, y are variable symbols, then x∩y is a term. We hope context will help in deciding whether x ∩ y is meant to be a term or a concrete set. (In the rest of this paper we tended to use ∧ and ∨ as operation symbols denoting the concrete operations ∩ and ∪. In the present proof however, we were afraid of potential confusion between ∧ as an algebraic operation and ∧ as a logical connective occurring in the language used for axiomatizing RCA_{ω}.) As before, x − y is the Boolean term (x ∩ −y) denoting the set x ∖ y. It is especially important to note that since, for the algebra 𝔄 constructed above, 𝔄 ∉ RCA_{ω} was not excluded, the operations denoted by ∪, ∩, c_{i} etc. in 𝔄 are not assumed to be the real, set theoretic ones. They are just abstract operations despite of the notation “∪” etc. The Boolean ordering on 𝔄 will be denoted by ≤.

CLAIM 1. 𝔄 ∉ RCA_{ω}.

Proof: For i, j < ω, i ≠ j, let s_{j}^{i}(x) = c_{i}(Id_{ij} ∩ x) and s_{i}^{i}(x) = x. Let the term τ be defined by

τ(x) =^{def} ⋂_{i≤m} s_{i}^{0}c_{1} … c_{m}x ∩ ⋂_{i<j≤m} −Id_{ij}.

Let ^{ω}U^{(q)} = {z ∈ ^{ω}U : {i ∈ ω : z_{i} ≠ q_{i}} is finite}. (^{ω}U^{(q)} is usually called the weak Cartesian space determined by U.) Then, for our concrete choice of R, 𝔄′ ⊧ τ(R) = 0, and this expresses that “the domain of R is of cardinality ≤ m”, because of the following:

c_{1} … c_{m}R = (U_{0} ×^{m}U × U_{m+1} × …) ∩^{ω}U^{(q)},

s_{i}^{0}c_{1} … c_{m}R = (^{i}U × U_{0} ×^{(m−i)}U × U_{m+1} × …) ∩^{ω}U^{(q)},

⋂_{i≤m} s_{i}^{0}c_{1} … c_{m}R = (^{(m+1)}U_{0} × U_{m+1} × …) ∩^{ω}U^{(q)} .

Then by |U_{0}| ≤ m we have that there is no repetition–free sequence in^{m+1}U_{0}. Thus 𝔄′ ⊧ τ(R) = 0.

Then 𝔄 ⊧ τ(R) = 0 by 𝔄′ ⊆ 𝔄. Assume that 𝔄 ∈ RCA_{ω}. Then there is a homomorphism h : 𝔄 → 𝔅 ∈ Cs_{ω} such that h(R) ≠ 0, for some 𝔅. By h(R) ≠ 0, there is s ∈ h(R). By R ≤ c_{0}R_{j} we have h(R) ⊆ c_{0}h(R_{j}), so there is u_{j} such that ⟨u_{j}, s_{1}, s_{2}, … s_{i} … ⟩ ∈ h(R_{j}), for all j ≤ m. These u_{j}’s are different from each other since the R_{j}’s are disjoint from each other, and so the h(R_{j})’s are disjoint from each other. Consider the sequence

z = ⟨u_{0}, u_{1}, …, u_{m}, s_{m+1}, …⟩.

Then z ∈ τ(R) is easily seen as follows. Obviously z ∈ −Id_{ij}, if i < j ≤ m. Further ⟨u_{i}, z_{1}, … z_{j} … ⟩ ∈ Id_{0i} ∩ c_{1} … c_{m}h(R), hence z ∈ s_{i}^{0}c_{1} … c_{m}h(R) if i ≤ m. A contradiction. QED(Claim 1)

CLAIM 2. The n–generated subalgebras of 𝔄 are in RCA_{ω}.

Proof: Let G ⊆ A, |G| ≤ n. For all i, j ≤ m define

R_{i} ≡ R_{j} iff (∀g ∈ G)[R_{i} ≤ g ↔ R_{j} ≤ g].

Then ≡ is an equivalence relation on {R_{j} : j ≤ m} which has ≤ 2^{n} blocks by |G| ≤ n. Let p denote the number of blocks of ≡, i.e. p = |{R_{j}/≡ : j ≤ m}| ≤ 2^{n} ≤ m. Define

B = {a ∈ A : (∀i, j ≤ m)([R_{i} ≡ R_{j} and R_{j} ≤ a] ⇒ R_{i} ≤ a)}.

We now show that B is closed under the operations of 𝔄.: Let k < l < ω.

1) B clearly is closed under the Boolean operations

2) Id_{kl} ∈ B since (∀j ≤ m)R_{j} ≰ Id_{kl}

3) Clearly, A′ ⊆ B (since R is an atom of 𝔄′), and c_{k}a ∈ A′ for all a ∈ A. Thus c_{k}b ∈ B (for all b ∈ A).

Let 𝔅 ⊆ 𝔄 be the subalgebra of 𝔄 with universe B. By G ⊆ B, it is enough to show that 𝔅 ∈ RCA_{ω}.

We will define an embedding h : 𝔅 ↣ ⟨𝔓(^{ω}U), c_{i}, Id_{ij}⟩_{i,j<ω}. Let {y_{j} : j < p} = {∑(R_{j}/ ≡) : j ≤ m}. Then {y_{j} : j < p} is a partition of R in 𝔅, c_{i}y_{j} = c_{i}R for all j < p and i < ω and every element of 𝔅 is a join of some element of A′ and of some y_{j}’s. So, 𝔅 looks like

FIGURE 3

We are going to define the images of the y_{j}’s. Let Q = {0, 1, …, m − 1} and let (Q, +, 0) be a group. For each i < ω let f_{i} : U_{i} ↣ → Q be a bijection such that f_{i}(q_{i}) = 0. For j < m define

R′′_{j} = {z ∈ R : ∑⟨f_{i}(z_{i}) : i < ω⟩ = j},

where ∑ denotes the group theoretic sum in (Q, +, 0). Then it is not difficult to check that the R′′_{j}’s are disjoint from each other and

c_{i}R′′_{j} = c_{i}R for all i < ω

for the concrete set theoretic c_{i}’s. Define for all j < p − 1

R′_{j} = R′′_{j}
R′_{p−1} = ⋃{R′′_{j} : p − 1 ≤ j < m}.

We are ready to define the embedding h of B.: We define for all b ∈ B

h(b) = (b − R) ∪ ⋃{R′_{j} : j < p, y_{j} ≤ b},

where b−R is computed in 𝔄, and since (b−R) ∈ A′, the rest of the operations are the concrete set theoretic ones. Now it is not difficult to check that h is an embedding h : 𝔅 ↣ ⟨𝔓(^{ω}U), c_{i}, Id_{ij}⟩_{i,j<ω} as follows.

Clearly h preserves ∪, −. h(b) = 0 implies b = 0, hence h is one–one. h(Id_{kl}) = Id_{kl}. Now we check c_{k}h(b) = h(c_{k}b).

c_{k}h(b) = c_{k}[(b − R) ∪ ⋃{R′_{j} : y_{j} ≤ b}]
= c_{k}(b − R) ∪ ⋃{c_{k}R′_{j} : y_{j} ≤ b}
= c_{k}(b − R) ∪ ⋃{c_{k}y_{j} : y_{j} ≤ b}
= c_{k}[(b − R) ∪ ⋃{y_{j} : y_{j} ≤ b}]
= c_{k}b,

where the operations in the first two lines are set theoretic while those in the last three lines are understood in the abstract algebra 𝔄. h(c_{k}b) = (c_{k}b − R) ∪ ⋃{R′_{j} : y_{j} ≤ c_{k}b} = c_{k}b, since (∃j)y_{j} ≤ c_{k}b iff R ≤ c_{k}b, and R ≰ c_{k}b iff c_{k}b = c_{k}b − R. QED(Claim 2) QED(Theorem 5.1)

The above proof of Thm 5.1 yields the following set E_{∞} of equations witnessing the nonaxiomatizability of (even the theory of) RCA_{ω} by finitely many variables. E_{∞} has the following properties:

RCA_{ω} ⊧ E_{∞}, but for any set Σ of universally quantified formulas containing only finitely many variables and valid in RCA_{ω}, we have

Σ |̸= E_{∞}.

Further, for any first order formula φ valid in RCA_{ω}, φ |̸= E_{∞}. The m-th equation e_{m} ∈ E_{∞} is the following:

⋂_{i≤m} c_{0}(x ∩ x_{i} ∩ ⋂_{i≠j≤m} −x_{j}) ≤ c_{0} … c_{m}( ⋂_{i<j≤m}(s_{i}^{0}c_{1} … c_{m}x) − d_{ij}).

E_{∞} = {e_{m} : m ∈ ω}, and as one expects, as m increases so does the number of variables occurring in e_{m}.

In [A91] it is carefully checked that E_{∞} indeed has the properties claimed above.

Remark 5.2 below describes the modifications needed for obtaining proofs for the analogous (with Thm 5.1) non-finitizability theorems for RCA_{n} (n > 2) and RRA. Along these modifications the above set E_{∞} of equations can be adapted to RCA_{n} and RRA, too. This is done in [A91]. REMARK 5.2: Here we outline the easy modifications of the above proof of Theorem 5.1 yielding simple proofs for Theorems 4, 2 (non-finite axiomatizability of RCA_{n} and RRA.)

We use the words “representable” and “representation” as introduced at the beginning of §3 and below the definition of ℜ𝔢𝔩_{n}(U) (this was in §4 above Theorem 3). More concretely, let α be n or ω. An algebra 𝔄 similar to RCA_{α}’s is said to be representable if 𝔄 ∈ RCA_{α}. As explained at the above mentioned two points of this paper, representability means that 𝔄 is isomorphic to an algebra 𝔄^{+} whose elements are α–ary relations (and whose greatest element is a disjoint union of Cartesian spaces). 𝔄^{+} is called a representation of 𝔄 and sometimes the isomorphism h : 𝔄 ↣ → 𝔄^{+} too is called the representation of 𝔄. By a homomorphic representation we understand a homomorphism mapping 𝔄 into some Cs_{α}. This concept receives its importance from the simple but useful fact that representability of 𝔄 is equivalent with the existence of a set H of homomorphic representation of 𝔄 such that (∀nonzero x ∈ A)(∃h ∈ H)h(x) ≠ 0.^{45}

The intuitive idea of the above proof of Theorem 5.1 was the following. We found two different ways of “counting” the elements of the domain {s_{0} : s ∈ R} of the relation R. (It is important that this counting was done by looking only at the abstract, i.e. isomorphism invariant properties of 𝔄). These ways of counting were: (1) Looking at the number of the disjoints elements R_{j} below R. This allowed us to conclude that the domain of R must be big. (2) Using the Id_{ij}’s exactly as one uses equality in first–order logic to express that a certain finite set is smaller than some m, we concluded that the domain of R must be small. (This was done by the term τ(R) in the proof of Claim 1.)

45For completeness it might be informative to mention here that the adjectives ‘representable’ and ‘nonrepresentable’ are especially useful when applied to members of the finitely axiomatizable variety CA_{n} properly containing RCA_{n}.

CA_{n} was discussed in an earlier subsection of §4. Nonrepresentable CA’s are, in a sense, nonstandard models of the axioms defining CA_{n}. For the present proof, though, we need not be aware of CA’s.

Then by splitting, we modified some 𝔄′ ∈ RCA_{ω} such that in the new algebra 𝔄 (1) and (2) would contradict each other ensuring 𝔄 ∉ RCA_{ω}. We were able to construct an infinite sequence 𝔄_{n} ∉ RCA_{ω} (n ∈ ω) of algebras this way, such that as n increases, the contradiction between (1) and (2) becomes weaker and weaker. Actually, as n approaches infinity, the contradiction betweeen (1) and (2) vanishes. So in the ultraproduct of the 𝔄_{n}’s, (1) and (2) do not contradict each other any more, and this ultraproduct is in RCA_{ω}. (To be precise, the final argument we used in the proof of Theorem 5.1 was not an ultraproduct construction. There the conflict between (1) and (2) became weaker and weaker in the sense that more and more elements of 𝔄 had to be inspected for discovering this contradiction. We find it more natural to explain the intuitive idea in terms of ultraproducts, which incidentally happens to be the way the original proof went.) This finishes the intuitive idea of the proof of Theorem 5.1.

Next we would like to repeat this proof for RCA_{n} in place of RCA_{ω} with 2 < n < ω. If we simply replace ω everywhere with n, the proof does not go through because the counting in (2) needs an arbitrarily large number of Id_{ij}’s and we have only n×n many. So we need a new method for doing (2), i.e. for looking at an abstract algebra 𝔄 together with its element R and concluding that in any (homomorphic) representation h : 𝔄 → 𝔅 ∈ Cs_{n} of 𝔄, the domain of h(R) must be of smaller size than a certain m. (The difficulty is that we have to be able to repeat this for arbitrarily large m ∈ ω.) There are many easy and natural ways for doing this (which will not interfere with the procedure of splitting used to induce a controllable contradiction between (2) and (1).^{46} One of them is the following. If we could “see” by looking at 𝔄 “abstractly” that U_{0} × U_{0} is a union of fewer than m functions f_{i} (i < m − 1) each of which is coded by an element of 𝔄, then the domain U_{0} of R must be of smaller cardinality than m in any representation of 𝔄. To simplify the construction, we choose these functions (the f_{i}’s) to be powers of a single suitable permutation f of U_{0}. (E.g., choosing U_{0} = m and f to be the usual successor modulo m works.) Let us fix such an f and let F =^{def} f ×^{n−2}U. Then F ⊆ ^{n}U. We include into our algebra 𝔄′, besides R, also F as a new generator element. Now, similarly to the way we used the equation τ(R) = 0 in the proof of Claim 1, by studying the abstract properties of F and R in the new 𝔄′ we can conclude that in any homomorphic representation h of 𝔄′, the Cartesian square of the domain of h(R) is contained in the union of fewer than m powers of a function coded by h(F). But then this domain must be of cardinality ≤ m. (Exactly what we proved in Claim 1 of the old proof. So we can prove our new Claim 1.) After this modification, the whole proof goes through by replacing all occurrences of ω with n.

This completes the outline of the proof that RCA_{n} cannot be axiomatized with quantifier free formulas using finitely many variables, if n > 2. Thus Theorem 4, too, is supplied with an outline of proof.

Let us turn to the RRA case, i.e. to Theorem 2. The idea is basically the same as in the above outlined RCA_{n} case. Exactly as in the RCA_{n} case, here too we use two counting principles (1), (2) and construct algebras 𝔄_{n} in which (1) and (2) contradict each other. Again we want a controllable contradiction such that as n approaches infinity the contradiction vanishes. We have to modify again the counting principle in (1) because in the RRA case functions interfere with splitting elements R = U_{0} × U_{1} (U_{0} ∩ U_{1} = 0) the way we did when constructing 𝔄_{n} = 𝔄 from 𝔄′ at the beginning of the proof of Theorem 5.1 (cf. Andréka–Maddux–Németi [AMdN]). The modification, however, is not as essential or “radical” as it was when we moved from RCA_{ω} to RCA_{n}. There are many counting principles one can use to implement (1) in the RRA case. E.g. we can use colorings of the full graph U_{0} × U_{0} with finitely many colors without monochromatic triangles, and then apply Ramsey’s theorem. This means that we arrange U_{0} × U_{0} to be a disjoint union of symmetric relations G_{0}, …, G_{r} such that G_{r} = Id ↾ U_{0} (symmetric means G_{i} = G_{i}^{−1}). Further we ensure that for each i, j, k < r, if |{i, j, k}| > 1, then G_{i} ∘ G_{j} ⊇ G_{k}, and (G_{i} ∘ G_{i}) ∩ G_{i} = 0. We let our 𝔄′ ∈ RRA be generated in this case by {G_{0}, …, G_{r}, R}. All these properties of the G_{i}’s were abstract, “equational” ones. This ensures that in every representation of 𝔄′, the domain U_{0} of R must be finite (by Ramsey’s theorem). We split R into ω many R_{i}’s obtaining 𝔄 from 𝔄′ as we did in the RCA_{ω}, RCA_{n} cases before. The rest of the proof goes through as before with replacing ω (or n) everywhere by 2, except for the following minor change. In the RRA case we have to look at the ultraproduct of the 𝔄_{n}’s and observe that it is representable (since the contradiction between (1) and (2) disappeared as both counting gives us continuum many elements). Therefore this proof gives only non-finite axiomatizability of RRA (i.e. Monk’s theorem) without proving (Jónsson’s result saying) that infinitely many variables are needed. For the latter one has to fine-tune the construction some more. From non-finite axiomatizability of RRA Theorem 2 easily follows since, as mentioned around the end of §3, RRA is easily seen to be finitely axiomatizable over BRA (it is enough to postulate that Id is the unit element of the semigroup “;” turning it into a monoid).

46This is something to keep in mind; when building new devices in our algebra in order to enable us to do (2), we do not want these new devices to hinder us in executing the rest of the proof the key step in which was the application of splitting producing an 𝔄 ∉ RCA_{ω} from the original 𝔄′ ∈ RCA_{ω}.

The proofs outlined in Remark 5.2 above are due to Andréka, cf. [A91]. In connection with the RRA-part of Remark 5.2 we should point out that Monk’s original proof also used graph colorings and Ramsey’s theorem for basically the same purpose, cf. [HMTII]. In this connection he initiated a new, fruitful research direction dealing with the connections between combinatorics and algebraic logic which was pursued further by Comer, Maddux, Andréka, Tuza and others.^{47} ([T91] is a recent source of references.)

Theorem 5.1 was proved for RCA_{α}, to be introduced soon, for any ordinal α > 2.^{48} In §2, Theorem 2 leads to Problem 1 in a natural way. Exactly the same way our present Theorem

5.1 leads to the following important open problem.

PROBLEM 2.0.1. Find \underline{simple}, mathematically transparent, decidable sets Σ of equations characterizing RCA_{ω}. (The RCA_{n}, 2 < n < ω version of this problem is open and interesting, too. This RCA_{n} version is strongly related to Problem 2.0 way above. On the other hand, the present, RCA_{ω} version has a logical counterpart similar to Problem 2.0 too, cf. e.g. [HMTII, Prob.4.16, p.180].) This is one of the central problems of Algebraic Logic, cf. [HMTII, Prob.4.1], Henkin–Monk [HM74, Prob.5] etc. For strongly related results (or for partial solutions) see [HMTII, pp.112–119], [V90, V91], [Si90, 91], [HH94].

Cf. also Problem 1 in §3 above and the remarks following and preceding that problem. Cf. also the remarks in §4 preceding the subsection entitled “The axiomatic… etc.”, and the last two pages of §4, especially Problems 2.1, 2.2.

Returning to the problem of finitization first discussed in Remark 2, the infinite number of basic operations of RCA_{ω} presents itself as a new kind of obstacle. Of course, one could try to alleviate this by changing the language of RCA_{ω} such that all the old operations {c_{i}, Id_{ij} : i, j ∈ ω} would become term definable in a new finite language and then one would search for a finite axiomatization. However, this approach to the problem belongs to a relatively big and central project, called true finitization, of mainstream algebraic logic. This project is in the spirit of Remark 2 way above, cf.

47This direction goes back to the pioneering work of Lyndon, but Lyndon used only a very special part (projective geometry) of combinatorics, while Monk generalized the approach to using combinatorics in general. This, and further historical information is in [HMTII], e.g. on p.101.

48Andréka found the above presented simple proof for Theorem 5.1 in January 1986, cf. Andréka [A91], Monk [M91].

This way she also found a simple proof for Monk’s important non–finitizability theorem ([HMTII, 4.1.7]).

Sain [S87a], and we will return to it at the end of this section, and in §§6,7. In the next paragraph, we will aim for axiomatizability with a finite schema only.^{49} We will need the following operations on relations, expanding RCA_{ω}. Let R ⊆ ^{ω}U and τ : ω ⟶ ω. Then S_{τ} (R) =^{def} {q ∈ ^{ω}U : q ∘ τ ∈ R}. The logical counterpart of the operation S_{τ} is substitution of variables along τ, e.g. S_{τ} (R_{1}(v_{0}v_{1})) is intuitively R_{1}(v_{τ(0)}, v_{τ(1)}). For completeness we note the following. Let [i/j] : ω ⟶ ω map i to j and leave all else fixed. Then S_{[i/j]} was already term-definable in RCA_{ω} and coincides with s_{j}^{i} defined in the proof of Theorem 5.1.

Advanced topics in RCA-theory

Now we can turn to \underline{finite schema axiomatizability} as a continuation of Remark 2. From now on, a finite schema is allowed to contain only finitely many variables. By Theorem 5.1, RCA_{ω} is not axiomatizable by a finite schema. W. Craig expanded RCA_{ω} with the two operations S_{succ} and S_{pred}, where the indices are the usual successor and predecessor functions on ω. Craig proved that the equational theory of the so expanded version of RCA_{ω} is axiomatizable by a finite schema. However, the price for this is that Craig’s algebras do not form a variety; in fact they are not closed under ultraproducts, so they do not form an axiomatizable class, cf. [HMTII, p. 265] or Craig [Cr74]. The most active proponents of Craig’s approach at present are, among others, I. Sain and R. J. Thompson, but see also the references on p. 265 of [HMTII], and §7 (2–4) herein. Continuing Craig’s work, Sain has obtained a class of algebras which is an adequate algebraic counterpart of first–order logic without equality. (In Sain’s algebras, for every finite transformation τ of ω, and for every i ∈ ω, S_{τ} and c_{i} are term definable, but Id_{ij} is not.) Sain proved that her variant of RCA_{ω} is a variety axiomatizable by a finite schema of equations.^{50} (This is related to but not identical with the results in the current version of Sain [S87a].) There are several intriguing open problems in Sain’s works on Craig’s approach, the solutions of which could provide considerable insight into the basic questions of algebraic logic. Most of these problems admit very simple purely semigroup theoretic formulations. (The main bulk of these is stated either in the June 1987 version or in the current version of Sain [S87a].)

Convention. Throughout, both ω and n ∈ ω are regarded as ordinals. Further, n = {0, …, n − 1} for n ∈ ω, and more generally α = {β : β is an ordinal and β < α} for any ordinal α. For any ordinal α, ^{α}U is the set of α-sequences of members of U. Formally, ^{α}U = {f : f is a function mapping α into U}. The definition remains the same if α stands for an arbitrary set (and is not necessarily an ordinal).

To treat RCA_{n} and RCA_{ω} in a unified manner, we replace ω in the definition of RCA_{ω} with an arbitrary but fixed ordinal α, obtaining RCA_{α} (here α = n and α = ω are of course permitted). This generalization will also be useful in algebraizing various quantifier logics different from classical first– order logic L_{ωω}. In particular, a full Cs_{α} is of the form ⟨𝔓(^{α}U), c_{i}, Id_{ij}⟩_{i,j<α} defined analogously to RCA_{n} and RCA_{ω}.

An important and natural generalization of RCA_{α} is obtained by relaxing the condition that the top (i.e. largest) element of a full Cs_{α} is of the form ^{α}U. Namely, let V ⊆ ^{α}U. By a full relativized Cs_{α}, or a full Crs_{α}, we understand ⟨𝔓(V), c_{i}, Id_{ij}⟩_{i,j<α}, where the operations c_{i} and Id_{ij} are now relativized to V, e.g. Id_{ij} = {q ∈ V : q_{i} = q_{j}}. (In the abbreviation Crs, the “r” refers to “\underline{r}elativized”.) Subalgebras of full Crs_{α}’s are called Crs_{α}’s, and the class of isomorphic copies of Crs_{α}’s is denoted by ICrs_{α} (see [HMTII, §5.5]). The elements of Crs_{α} are still natural algebras of α–ary relations. The elements of a Crs_{α} 𝔄 are obviously α–ary relations, the only difference with RCA_{α}’s is that now the greatest relation (i.e. the top element of 𝔄) is an arbitrary α–ary relation and not necessarily a Cartesian space or a disjoint union of such spaces. This flexibility will be useful in algebraizing quantifier logics different from L_{ωω} (like higher order, many-sorted, and nonclassical logics); cf. e.g. [N78]. Before looking into different quantifier logics, we mention some algebraic results.

49Here finite schema means something like (E0–7) in §8 (Appendix) herein. There are precise definitions of a finite schema in [HMTII, pp. 110, 261] but it is not necessary to look these up in order to understand the present paper; it is enough to know that a finite schema may contain only finitely many variables, in the sense of [HMTII].

50This result appears as Thm.1 on p. 3 of the June 1987 version of Sain [S87a] (a version of [S87a] that differs substantially from the current one). Thm.1 is proved on pp. 34–37 in §2, and Remark 10.1 on p. 37 contains a simplified finite schema defining her version of RCA_{ω}. This schema is indeed simple. Page and item numbers in this footnote refer to parts of op. cit.

THEOREM 6.

(i) ICrs_{α} is a variety whose equational theory is not axiomatizable by a finite schema (for α > 2).

(ii) ICrs_{α} is \underline{decidable}. Moreover, the set of universally quantified formulas valid in ICrs_{α} is

decidable, too. (iii) A finite Crs_{n}, for finite n, is isomorphic to a Crs_{n} with finite greatest element.

For a proof of (i) see [HMTII, Thm’s 5.5.10, 5.5.13]. For proof of (ii) see [N86] and [N96a], [N96b]. For proof of (iii) see Andréka–Benthem–Németi [AvBN96] and Andréka–Hodkinson–Németi [AHN96].

Related results for RRA’s are in Maddux [Ma82], Németi [N87a] (decidability for relativized RRA’s), Marx [Ma93]; see also Kramer [K89], [Ma93a], [MMNS], [MP92], [Ma94], [AHN96]. The equational theory of BRA is already undecidable, moreover so is that of any variety containing as a subreduct either the Id-free subreduct of RCA_{3} or the ^{−1}-free subreduct of BRA (a result of Maddux; see [HMTII, 5.1.66]). As a contrast, the equational theory of RCA_{2} is both decidable and finitely axiomatizable ([HMTII, Thm’s 4.2.9, 3.2.65 pp. 136, 84]).

The following is in contrast with Theorems 5.1 and 6(i). There is an easily understandable set Σ of equations using only one variable in the extra-Boolean equations such that ICrs_{α} = Mod(Σ). This Σ is due to Resek, Thompson, and was polished by Andréka. See Monk [M91, Thm.9.4] for Σ and a short proof due to Andréka. Studying this result ([M91, Thm.9.4]) might illuminate one of the most important open problems of Algebraic Logic, which is Problem 2.0.1 above. How a positive solution using schemas might be possible in spite of Theorem 5.1 (and the radical improvements of Theorems 4, 5.1 in Andréka [A91]) is illustrated by [M91, Thm.9.4], Sain–Gyuris [SGy94] Thm.2.5, and [HH94].

An important positive result related to the finitization problem for algebras of relations (see Remark 2, and the discussion not too far above) is based on the following breakthrough by Diane Resek. In the sequel, 1 is the Boolean constant denoting the top element of the algebra.

THEOREM 7 (Resek–Thompson). The elements of Crs_{α} satisfying c_{i}Id_{ij} = 1 (for all i, j ∈ α) form a variety V_{α} axiomatizable by a finite schema of equations. (If α is finite, V_{α} is finitely axiomatizable.)

Note that a Crs_{α} 𝔄 satisfies c_{i}Id_{ij} = 1 iff its top element satisfies a very simple set theoretic condition^{51}. Further interesting results and proofs in connection with relativization of cylindric algebras (in particular of Cs_{α}’s) are in Maddux [Ma89d]. The possibility of proving Theorem 7 from the Ultrafilter Existence Axiom only instead of using the full power of the Axiom of Choice is also discussed in [Ma89d]. (Further, [Ma89d] contains improvements and discussions of some of the conditions in [AT88].) We note that [Ma89d] contains a strengthening of Theorem 7 going in a direction called “relativization” not discussed in the present paper.^{52}

51The original proof (of Theorem 7) was very long, see [HMTII, 3.2.88], p.101. The first short proof appeared in [AT88]. (Andréka used her method later for proving further results, cf. e.g. [A91, Prop.3, p.64], the proof of [M91.

Thm.9.4] discussed above. A. Simon obtained Theorems 8/1 and 8 below, together with further results in this line, also by refining the method of Andréka.)

The subvariety V_{α} of ICrs_{α} in Theorem 7 has a decidable equational theory if α < ω (Németi [N86]). It is an open problem whether this carries over to α = ω.

Let Dfrs_{α} be the class of subalgebras of Id_{ij}–free reducts of Crs_{α}’s. I.e. a Dfrs_{α} is an algebra 𝔄 ⊆ ⟨𝔓(V), c_{i}⟩_{i<α} with V ⊆ ^{α}U for some V and U. Dfrs abbreviates “\underline{d}iagonal–\underline{f}ree cylindric– \underline{r}elativized \underline{s}et algebra” (the relation Id_{n} is often called the “diagonal” of the space ^{n}U). The following is in contrast with Theorem 6 above.

THEOREM 8 (A. Simon). Dfrs_{α} is axiomatizable by a simple finite schema of equations. Actually, axioms (E0–E3) listed in §8 (in the proof of Proposition 8.3), close to the end of this work, provide a complete and sound axiomatization for Dfrs_{α}.

Simon extended the above theorem to the expansions of Dfrs_{α}’s with the operations C_{(Γ)} with Γ ⊆ α where C_{(Γ)} will be defined very soon. (Our definition coincides with that of [HMTII].) The equational theory of the expanded algebras remains decidable (a result of Andréka). To study Dfrs_{α} + C_{(Γ)} was suggested by S. Comer and D. Vakarelov independently, both with motivations from computer science (e.g. data base theory).

Let X ⊆ V ⊆ ^{α}U and Γ ⊆ α. Then we define

C_{(Γ)}^{V}(X) =^{def} C_{(Γ)}(X) =^{def} {q ∈ V : (∃s ∈ X) (∀i ∈ α ∖ Γ) q_{i} = s_{i}} .

By a strong Dfrs_{α} (diagonal–free strong cylindric–relativized set algebra) we mean a subdirect product of algebras of the form ⟨𝔓(V), C_{(Γ)}^{V} : Γ ⊆ α⟩ with V ⊆ ^{α}U for arbitrary V and U.

THEOREM 8/1 (A. Simon). Let α be arbitrary. The class of strong Dfrs_{α}’s is a variety axiomatizable by the following equations (S0 − S5), postulated for each Γ, Δ ⊆ α. (S0) The Boolean equations (S1) C_{(Γ)}(x ∨ y) = C_{(Γ)}x ∨ C_{(Γ)}y (S2) x ≤ C_{(Γ)}x = C_{(Γ)}C_{(Γ)}x (S3) −C_{(Γ)}x = C_{(Γ)} − C_{(Γ)}x (S4) C_{(Γ)}x ≤ C_{(Δ)}x if Γ ⊆ Δ ⊆ α (S5) C_{(∅)}x = x.

Note that (S0 − S3) say simply that C_{(Γ)} is a complemented closure operator (of some BAO). The proof uses methods analogous with Andréka’s proof ([M91, proof of Thm.9.4]), of the

Resek–Thomson equational axiomatization of ICrs_{α}, mentioned between Theorems 6, 7 above, cf. also [AT88].

Theorem 8/1 is a special case of a more general theorem about Q − Dfrs_{α}’s, where Q ⊆ P(α) is arbitrary but fixed, and a Q−Dfrs_{α} is a subdirect product of algebras of the form ⟨𝔓(V), C_{(Γ)}^{V}⟩_{Γ∈Q}. Of particular interest is the choice of Q_{0} = all finite subsets of α. Note that the operation C_{({0,1})} is not term definable in Crs_{α}’s. (An observation of Dimiter Vakarelov.) More detail and proofs are available from A. Simon (for address cf. [Si90]).

Theorem 8/1 above improves [Co91a, Thm.1.6]. Namely, Theorem 8/1 gives a natural geometric or “relational” representation for all members of the variety generated by the so called knowledge representation algebras of [Co91a] coming from comp. sci. (together with a transparent finite axiomatization of this variety). It is outlined in [Co91a] how the algebras in Theorem 8/1 are important for information systems. Further computer science connections of Theorem 8/1 were indicated in Düntsch [Dü91a] (where, roughly speaking, the data–base theoretic projection operator Π_{Γ} is defined by Π_{Γ}(x) =^{def} C_{(α∖Γ)}(x), for all Γ ⊆ α).

52I.e.: [Ma89d] contains a theorem from which the present Thm.7 can be derived (but not vice versa). However, this (interesting cylindric algebraic) theorem of [Ma89d] lies outside the scope of the present work, hence it is not discussed here.

Let us turn to applying Crs_{α} to \underline{algebraizations of different quantifier logics}^{53}.

We will discuss here only one of the many applications of Crs_{α} to logic. For the rest of such applications we refer to e.g. [vB96], [AvBN96], [AvBN96a], [Al95], [N96a], [VM96].

Let us consider many-sorted logic first (as shown in Barwise–Feferman [BF85] or [S87b], many– sorted logic is a good unifying framework for abstract model theoretic logics, nonclassical, and various unusual logics). Let U and W be two disjoint sorts in the logic we want to algebraize. Let V = ^{ω}U ×^{ω}W. Then, according to our Convention concerning associativity of × and Cartesian power, V ⊆ ^{ω+ω}(U ∪^{.} W). Then 𝔄 = ⟨𝔓(V), c_{i} …⟩_{i,j<ω+ω} is a Crs_{ω+ω}. Here c_{i} corresponds to quantifying over variables of sort U or W, depending on whether or not i < ω, and similarly for Id_{ij} when i and j are “in the same copy of ω”. The fact that Id_{0ω} = 0 corresponds to the logical fact that equality is not defined between elements of different sorts. Now, we could repeat, with appropriate changes, what we said before about the connection between RCA_{ω} and L_{ωω}. The idea extends to an arbitrary number of sorts, in the obvious way. See Németi [N78], LeBlanc [L59, L62], [AGN75], Plotkin [Pl89, Pl88], for more detail. Now, many-sorted logic is a natural stepping stone toward higher order logic.

To treat second–order logic, for example, we can take V = ^{ω}U ×^{ω}P(U) and expand the Crs_{ω+ω} ⟨𝔓(V), c_{i} …⟩ with the new constant E = \{q ∈ V : q_{0} ∈ q_{ω}\} to represent algebraically the “\underline{e}lement of” relationship between the first–order and second–order objects. One can do the same for higher order logics, see e.g. [AGN75, AGN75a], LeBlanc [L59], Venne [V65, V66], Salibra [Sa89].

It is also indicated in Németi [N78], Freeman [F76], Georgescu [G79a], Monk [M60] how Crs_{α} can be used for algebraizing nonclassical quantifier logics like first–order modal logic or first–order temporal logic. The use of Crs_{α}’s as a unifying framework is more explicit in the first citation, while more details are available in the others, especially in [F76]. Roughly speaking, a Kripke model for first–order modal logic consists of a universe U of individuals, and a set W of possible states or “worlds” or time instances together with some further structure. In this case we choose the top element V of our Crs_{α} to be W ×^{ω}U. Intuitively, ⟨w, q_{0}, …, q_{i}, …⟩ ∈ R means that ⟨q_{0} … q_{i} …⟩ is in the relation R when viewed from the possible world or state w. The point we are trying to make here is that algebraizing non-classical (say modal) first–order logics again leads naturally to an algebra of finitary relations. The choice of the logic introduces some peculiarities into the relations, e.g. in the case of modal or temporal logics the first argument of the relations comes from a different “universe” (namely, that of possible states or time instances) than the rest of the arguments. This, however, does not seem to diminish the importance of the fact that we are dealing with relations. This is why Crs_{α} seems to be a reasonable unifying framework for the algebraizations of these logics.

∗ ∗ ∗ ∗ ∗

Let us return briefly to RCA_{ω}, and to the quest for finding natural algebras of finitary relations (we mean the quest we tried to illustrate by the “obvious“ considerations in connection with P(^{<ω}U) in §2, and which admitted one possible satisfactory solution by our defining ℜ𝔣(U), and the class L below Theorem 5).

Let Frl(U) = \{R : R ⊆ ^{n}U for some n ∈ ω\}. That is, Frl(U) is the collection of all \underline{f}initary \underline{r}e\underline{l}ations over the set U. A relation R ⊆ ^{n}U is essential if either R = ∅ or for no S ⊆ ^{(n−1)}U is R = S × U. So, an n–ary relation is essential if it is not a “dummy embedding into rank n” of another relation of some smaller rank. We let Ref (U) = {R ∈ Frl(U) : R is essential}. We define a partial ordering ≤ on Frl(U) as follows: For any R, S ∈ Frl(U) we let R ≤ S iff either S ⊆ R ×^{k}U, or S ×^{k}U ⊆ R for some k ∈ ω (i.e. if we level out the rank differences between S and R by using “dummy embedding” then S becomes a subset of R). Note that for all R, ∅ ≤ R and R ≤ ^{0}U(= {∅} = {⟨⟩}) because R ⊆ ^{n}U = ^{0}U ×^{n}U for some n.

53Further investigations of Crs_{α} and related classes as well as some of their applications in logic can be found in Alechina [Al95], Andréka-Goldblatt-Németi [AGN96], Mikulás [Mi95, Mi96], Marx [Ma94], [MMN95].

FACT 2. ≤ is a complemented distributive lattice ordering on Ref (U).

We let ⟨Ref (U), ∧, ∨, −⟩ be the Boolean algebra induced by this ordering ≤. Note that to every R ∈ Frl(U), there is a smallest (according to ≤) upper bound ess(R) of R in Ref (U). For R ⊆ ^{n}U and i < n let c_{i}(R) ⊆ ^{n}U be defined as in RCA_{n}. Let R ∈ Ref (U) and i < n. Then c_{i}^{+}(R) = ess(c_{i}(R)). For i ≥ n we let c_{i}^{+}(R) = R. Finally, for j < i, let I_{ij} = I_{ji} = {q ∈ ^{i+1}U : q_{i} = q_{j}}. So in particular I_{01} = {⟨a, a⟩ : a ∈ U}, I_{12} = U × I_{01}, and I_{n,n+1} = ^{n}U × I_{01}. Now, ℜ𝔢𝔣(U) = ⟨Ref (U), ∧, ∨, −, c_{i}^{+}, I_{ij}⟩_{i,j<ω}.

FACT 3.

(i) ℜ𝔢𝔣(U) ∈ RCA_{ω}, moreover RCA_{ω} is the variety generated by the class K = {ℜ𝔢𝔣(U) : U is a set}.

(ii) For L as introduced below Theorem 5, we have K = L up to isomorphisms, because ℜ𝔢𝔣(U) ≅ ℜ𝔣(U).

The above ideas are described in more detail in Andréka [A77, §V and p. 37]. We note that ℜ𝔢𝔣(U) seems to be related to Craig’s algebras of sets of finite sequences, with universe P(^{<ω}U), in §7.(3,4) below and in [HMTII, p. 265]. The nature of the relationship is not clear to us however (except that Ref (U) ⊆ P(^{<ω}U)). We included the algebras ℜ𝔢𝔣(U) here to show that the generators of the variety RCA_{ω} can be built up as algebras whose elements are truly finitary relations in the most concrete possible sense. Further the extra–Boolean operator c_{i}^{+} when applied to an (i+1)–ary relation R results in literally deleting the “last column” of R, which is one of the most natural and most widely used operations on n–ary relations. E.g. c_{2}^{+}{⟨a, b, c⟩} = {⟨a, b⟩}.

∗ ∗ ∗ ∗ ∗

As we mentioned above Theorem 6, in our first extension of Remark 2 (finitization) to RCA_{ω}’s i.e. in our discussion of the search for finite schema axiomatizable expansions of RCA_{ω}’s, there exists a more ambitious approach aiming at \underline{true finitization} of expansions of RCA_{ω}’s and their variants, see e.g. Monk [M70], Henkin–Monk [HM74, Problem 1], and Sain [S87a]. Roughly speaking, we are searching for an expansion V_{1} of RCA_{ω}’s such that V_{1} would be term definably equivalent (polynomially equivalent in sense of [HMTI, p. 125]) to a finitely axiomatizable variety V_{2}. This search is one incarnation of the general question, playing a central rôle in algebraic logic, which asks whether the choice of the fundamental operations of the algebras RCA_{ω} of relations was the best possible, or whether a different choice of the fundamental operations could improve the situation. Certainly we made some very quick decisions when passing from RRA’s to RCA_{n}’s, e.g. we decided to drop “∘” and ^{−1} from the list of fundamental operations and derive them as term functions. We will return to the issue of these decisions in general in §7, but the subject of finitization that recurs throughout the paper is also an investigation of possible alternatives to these decisions. Next we briefly return to finitization.

Generalizing our definition that preceded Problem 2, we now define, for an arbitrary ordinal α, an operation f : P(^{α}U) ⟶ P(^{α}U) to be permutation invariant iff for any permutation p of U and R ⊆ ^{α}U, we have p(f(R)) = f(p(R)). Consider the following statement. (4.1) There are permutation invariant operations f_{i} : P(^{ω}U) ⟶ P(^{ω}U), (i ≤ n) for each set U,

such that, if 𝔑(U) = ⟨𝔓(^{ω}U), f_{0}, …, f_{n}⟩ and K^{+} = {𝔑(U) : U is a set}, then the following holds. The RCA_{ω} operations c_{i} and Id_{ij} are term definable in K^{+}, i.e., there are terms γ_{i} and δ_{ij} (i, j ∈ ω) of K^{+} such that in 𝔑(U), γ_{i} and δ_{ij} define the usual c_{i} and Id_{ij}.

Note that (4.1) implies RCA_{ω} = SPRdK^{+}.

PROBLEM 2.1. Is (4.1) above true in such a strong form that for K^{+} in it, SP(K^{+}) is a finitely axiomatizable variety or at least such a quasivariety?

Problem 2.1 is the RCA_{ω} counterpart of the stronger, (3.2)^{+}-part of Problem 2 in Remark 2^{54}. The variant of Problem 2.1 corresponding to the slightly weaker, (3.2)-part of Problem 2 was solved by Sain (compare [S87a]), who proved the following.

THEOREM 8/2 (Sain). (4.1) is true in such a strengthened form that K^{+} in it generates a finitely axiomatizable variety.

This result indicates that there really is a difference between the two versions of Problem 2. Further, Sain exhibited easier (than the proof of Theorem 8/2) solutions for the counterpart of the weaker variant of Problem 2 quoted from [TG87] at the end of Remark 2. Namely, she provided easier proofs for the following:

PROPOSITION 8/3. (4.1) is true in such a form that to K^{+} in it there is a finitely axiomatizable variety K_{0} ⊇ K^{+} such that RCA_{ω} = SRdK_{0}. Furthermore, Rd is taken using the terms γ_{i} and δ_{ij} fixed in (4.1).

Sain also suggested several attractive simple choices of K^{+} and K_{0}, and raised the problem whether these choices also satisfy the statement of Proposition 8/3. (The only problem she left open is to check if the seven cylindric postulates, like c_{i}c_{j} = c_{j}c_{i}, in [HMT, p. 162] are satisfied in K_{0}.)^{55} Compare the June 1987 version of [S87a].

The fact that Sain’s proof for Proposition 8/3 is easier than her proof for Theorem 8/2 points in the direction that the variant of Problem 2 at the end of Remark 2 might turn out to be easier (than even the (3.2)-part of the original Problem 2). A finite schema version of Proposition 8/3 above was obtained by Craig before Sain proved her result, cf. Craig [Cr74] and our discussion of finite schema axiomatizability above Theorem 6.

PROBLEM 2.2. What is the answer to Problem 2.1 if we replace each occurrence of ω with n (for some n > 2) in it? This means replacing RCA_{ω}, ^{ω}U with RCA_{n}, ^{n}U respectively everywhere in Problem 2.1 and in item (4.1). This amounts to asking the same question that was asked in Remark 2 but now for RCA_{n} instead of BRA.

We will return to finitization in §6 beginning with Theorem 16, and also in §7 (9). (However, a rather strong negative finitization result, due to Andréka, will be mentioned in a footnote at the beginning of §6.)

§5. Algebras for logics without equality

So far we have extended RRA’s (with extra–Boolean operations ∘, ^{−1}, Id) to algebras of relations of higher ranks. Let us see how to extend BRA’s whose extra–Boolean operations were only ∘ and −1 to higher ranks. When extending RRA’s to higher ranks, we were able to drop the algebraic counterpart ^{−1} of substitution because by using c_{i} and Id one can express ^{−1} as a derived operation. (This was shown in the paragraph discussing the rôle of RCA_{ω} in the algebraization of first–order logic in the remarks following Theorem 5.) But when extending BRA’s we do not have Id; hence we cannot express ^{−1}, and thus shouldn’t drop it. To discuss the counterpart of ^{−1} for n–ary relations, we will use the operations S_{τ} : P(^{ω}U) ⟶ P(^{ω}U) for τ : ω ⟶ ω introduced in §4 immediately above the subtitle “Advance topics in RCA–theory”. We use the notation n = {0, …, n − 1}. Let n be fixed, and let i, j < n. Then [i, j] : n ↣ → n denotes the permutation of n interchanging i and j, and leaving everything else fixed. [i/j] : n ⟶ n sends i to j and leaves all else fixed. (To be unambiguous we should write [i, j]_{n} and [i/j]_{n} but we rely on context to make things clear.) For τ ∈ ^{n}n, S_{τ} : P(^{n}U) ⟶ P(^{n}U) is defined the same way as it was for the τ ∈ ^{ω}ω case. We let p_{ij} =^{def} S_{[i,j]} and s_{j}^{i} =^{def} S_{[i/j]}. So in particular p_{01}({⟨a, b, c …⟩}) = {⟨b, a, c …⟩} or more generally p_{01}(R) = {⟨q_{1}q_{0}q_{2}q_{3} …⟩ : ⟨q_{0}q_{1}q_{2} …⟩ ∈ R}. Clearly for n = 2 we have R^{−1} = p_{01}(R), and hence we may consider p_{ij} as (an economical) generalization of ^{−1} to higher ranks. The logical counterpart of p_{01} is the “substitution” sending the formula R(v_{0}v_{1}v_{2} …) to R(v_{1}v_{0}v_{2} …). If we want to algebraize first–order logic, we also need the substitution sending R(v_{0}v_{1}) to R(v_{1}v_{1}). Strangely enough, we cannot express this by using the p_{ij}’s and the c_{i}’s in themselves. Therefore we also need s_{1}^{0}. We could stop at this point, but for aesthetic reasons we add all the s_{j}^{i}’s for i, j < n, and not only s_{1}^{0}. These are not essential however, and the reader is invited to develop his/her version of algebraic logic having only one of the s_{j}^{i}’s.

54Problem 2.1 is a concretized instance of the similar one in Monk [M70] and Henkin–Monk [HM74] mentioned in Remark 2. It is closer to the problem in these papers than Problem 2 in Remark 2 was.

55That this might be a nontrivial task is indicated by Corollary 3.6 on p. 654 of Demaree [D72], as will be discussed below Theorem 16 herein.

The full polyadic set algebra of n–ary relations (the full Ps_{n}) over U is defined to be

⟨𝔓(^{n}U), c_{i}, s_{j}^{i}, p_{ij}⟩_{i,j<n}.

By a representable polyadic algebra of n–ary relations (an RPA_{n}) we understand a subalgebra of a direct product of full Ps_{n}’s up to isomorphisms. Formally,

RPA_{n} = SP(full Ps_{n}).

(Cf. the sentence above Remark 1 in §3.)

THEOREM 9. RPA_{n} is a variety axiomatizable by a decidable set of equations.

The proof of Theorem 1 also proves that RPA_{n} is a variety. See also the references in §5.4 of [HMTII]. The second part of Theorem 9 is proved by the methods in [HMTII, pp. 112–118].

THEOREM 10 (Johnson [J69]). RPA_{n} is not finitely axiomatizable if n > 2.

This result can, perhaps, be improved, since Andréka has a construction that might be suitable for proving that RPA_{n}, for n > 2, is not finitely axiomatizable over its p_{ij}–free subreduct, i.e. over the variety^{56} generated by the ⟨𝔓(^{n}U), c_{i}, s_{j}^{i}⟩_{i,j<n}’s. It would be interesting to know whether this improvement of Theorem 10 is true. This amounts to asking whether the equality–free (i.e. RPA– theoretic) version of Theorem 11 (i) below holds. It would also be interesting to know if Andréka’s generalization of Theorem 4 (non–axiomatizability by finitely many variables) carries over to RPA_{n}. (By Theorem 11 (iii), it does carry over to RPEA_{n}.)

We can put cylindric algebras and polyadic algebras together obtaining the so–called RPEA_{n}’s (\underline{r}epresentable \underline{p}olyadic \underline{e}quality \underline{a}lgebras), and we can ask ourselves if we get essentially more than RCA_{n}’s. In a sense, Andréka gave an affirmative answer, solving Problem 5.8 of [HMTII]. We give more detail: RPEA_{n} is obtained by expanding RPA_{n}’s by the constants Id_{ij} of RCA_{n}’s. Then RCA_{n}’s become subalgebras of reducts of RPEA_{n}’s. By the proof of Theorem 1, RPEA_{n} is a variety.

56These are Pinter’s substitution–cylindric algebras (of n–ary relations) (RSC_{n}’s); they will recur a few more times below. Andréka also proved that RCA_{n}, for n > 2, is not finitely axiomatizable over RSC_{n} either (s_{j}^{i} is an RCA_{n}– term; hence RSC_{n}’s are also subreducts of RCA_{n}’s). Andréka systematically generalized these kinds of results of hers to schemas for α ≥ ω, in analogy with Monk’s non–finitizability result for RCA_{ω} quoted from [HMTII, 4.1.7] above Theorem 6. She further strengthened most of these to stating non–axiomatizability by any set Σ of universally quantified sentences such that Σ contains only finitely many variables. This way she does not have to introduce schemata.

THEOREM 11 (Andréka–Tuza, Andréka).

(i) RPEA_{n} is not finitely axiomatizable over RCA_{n} for n > 2.

(ii) RPEA_{n} is not finitely axiomatizable over RPA_{n} either, for n > 2.

(iii) Both (i) and (ii) remain true if we replace finite axiomatizability by axiomatizability with a set Σ of universally quantified formulas such that Σ uses only finitely many variables.

For (i) see Andréka–Tuza [ATu88]. For the rest of Theorem 11 see Andréka [A91].

(ii) above is in contrast with the finite axiomatizability of RRA’s over BRA’s. Furthermore, Theorem 11 (i) is in contrast with the finite axiomatizability of RRA over its ^{−1}–free subreduct mentioned just before beginning of §4.

In analogy with the relationship between RCA_{n}’s and RRA’s, BRA’s are subreducts^{57} of reducts of RPA_{n}’s, for n > 2. In particular, if R, S ⊆ U × U and if Dr(R) = R × U ⊆ ^{3}U, then Dr(R ∘ S) = c_{2}[p_{12}(Dr(R)) ∩ p_{02}(Dr(S))]. So, analogously to the remark following our definition of RCA_{n}’s, we can observe that the algebras of relations of higher ranks without equality (RPA_{n}’s) are expansions of the analogous algebras of binary relations (BRA’s). We already noted in §4 that RCA_{2}’s are subreducts of RRA’s. So RRA’s are in between RCA_{3}’s and RCA_{2}’s. (When algebraizing proof theory as opposed to the present emphasis on model theory — i.e. when dealing with the “axiomatic” classes CA_{n} and RA approximating RCA_{n} and RRA, discussed in the middle of §4 and in §8 — these numbers increase by one, so instead of 2 and 3, we get 3 and 4; see Maddux [Ma78, Ma83], [HMTII, §5.3], [TG87].) Strangely enough, RPA_{2}’s are not subreducts of BRA’s because s_{1}^{0} is not a derived operation of BRA’s. It would be nice to see a theory of BRA expanded with s_{1}^{0}. (Probably it would be somewhere halfway in between BRA’s and RRA’s.)

In RPA_{n}, the operation S_{τ} : P(^{n}U) ⟶ P(^{n}U) is term definable, for each τ : n ⟶ n. These S_{τ} ’s are quite important in RPA_{n}–theory. In this connection, recall from the subsection in the middle of §4 (cf. also §8) that in the literature there is a traditional, finitely axiomatizable variety PA_{n} of n–ary polyadic algebras approximating RPA_{n}. The classical axiomatizations of PA_{n} use all the S_{τ} ’s and not only the s_{j}^{i}’s and p_{ij}’s, see [HMTII, Def.5.4.1, p. 225]. Therefore the universal algebraic approach in [HMTII, pp. 260–263] based on the concept of a finite schema as defined in [HMTII] was not applicable to PA–theory. However, in Sain–Thompson [ST91], a single natural finite schema Σ_{0} of equations was found, such that Σ_{0} involves only the s_{j}^{i}’s and p_{ij}’s (and does not involve the S_{τ} ’s in general) besides the Boolean operations. It is proved there that PA_{n} = Mod(Σ_{0}). In particular, the traditional polyadic axioms like S_{τ∘σ}(x) = (S_{τ} ∘ S_{σ})(x) = S_{σ}(S_{τ} (x)) for all τ, σ ∈ ^{n}n follow from Σ_{0}. In other words, Σ_{0} is equivalent with the traditional axiomatization of PA_{n}, and at the same time Σ_{0} has certain advantages. An important point here is that the same Σ_{0} works for all n (and even for “n = ω” but that comes later). The key part of their Σ_{0} consists of Jónsson’s seven defining relations in [HMTII, 3.2.17(B), p. 68] for the semigroup ^{n}n as generated by the [i, j]’s and [i/j]’s transcribed to the p_{ij}’s and s_{j}^{i}’s. E.g. Jónsson’s defining relation [i, j] ∘ [i, j] = Id translates to the axiom p_{ij}p_{ij}(x) = x in Σ_{0}.

From now on, we will freely use the derived operations or term functions S_{τ} of RPA_{n}’s, for τ ∈ ^{n}n, without recalling them.

Generalizing RPA_{n} to RPA_{α} with α an arbitrary ordinal (especially for α = ω) goes exactly as in the RCA_{α} case. However, for historical reasons, the resulting algebras are called quasi–polyadic algebras instead of polyadic algebras (though it was these algebras which were introduced first, cf. Halmos [Ha54], and originally they were called polyadic; but later the terminology changed). In particular, for finite α, polyadic and quasi–polyadic algebras coincide^{58}.

⟨𝔓(^{α}U), c_{i}, s_{j}^{i}, p_{ij}⟩_{i,j<α} is called the full quasi–polyadic set algebra (full Qps_{α}) of α–ary relations over U for any ordinal α and set U. A quasi–polyadic set algebra (a Qps_{α}) is a subalgebra of a full Qps_{α}. Let α ≥ ω. Then by a representable quasi–polyadic algebra of α–ary relations (an RQA_{α}) we understand an isomorphic copy of a Qps_{α}. (Note the difference with the previous definitions!)^{59} RQA_{α} = IQps_{α}, where I is the operator of taking isomorphic copies. We define RQA_{n} = RPA_{n}, for n ∈ ω.

57Recall our Convention that subreducts are subalgebras of generalized reducts.

58The distinction between polyadic and quasi–polyadic will be useful later at the algebraization of infinitary logics.

THEOREM 12.

(i) RQA_{α} forms a variety if α ≥ ω.

(ii) For countable α, RQA_{α} is axiomatizable by a decidable set of equations.

The weaker form of Theorem 12 (i) saying that SPRQA_{α} is a variety is proved by repeating the proof of Theorem 5 with the obvious changes.

Note the striking difference with cylindric algebras: by Theorem 12 (i), IQps_{α} is a variety for α ≥ ω, but ICs_{α} is not. This result was perhaps known earlier, but the proof can be found in Sain [S89, Thm.3].

τ ∈ ^{ω}ω is called a finite transformation of ω iff {i ∈ ω : τ(i) ≠ i} is finite. For every finite transformation τ of α, the operation S_{τ} is term definable in RQA_{α}.

In connection with the elegant finite schema Σ_{0} of equations in Sain–Thompson [ST89] mentioned between Theorems 11 and 12 above, we note that Σ_{0} works for α ≥ ω; namely Σ_{0} implies all the distinguished quasi–polyadic axioms like S_{τ∘σ} = S_{τ} ∘ S_{σ} for finite transformations τ, σ of α. For a list of all these axioms see [HMTII, item 9, p. 266]; cf. also [HMTII, Def. 5.4.1]. I.e. for α ≥ ω, QPA_{α} = Mod(Σ_{0}), where QPA_{α} is the finitizable variety of α–ary quasi–polyadic algebras approximating RQA_{α} in the sense outlined in the middle of §4 and in §8.

For any finite subset Γ = {i_{1}, …, i_{n}} of α we can introduce the derived operation (or term function) C_{(Γ)}(x) = c_{i_{1}} … c_{i_{n}}(x). To see that this works, it is enough to recall that c_{i}c_{j} = c_{j}c_{i} is valid. An equivalent and frequently used way of introducing RQA_{α}’s is to start out from the algebras

⟨𝔓(^{α}U), C_{(Γ)}, S_{τ} : Γ ⊆ α is finite, and τ ∈ ^{α}α is a finite transformation⟩.

Indeed, this approach was taken in [HMTII, p. 266], Halmos [Ha62], [DM63], Andréka–Gergely– Németi [AGN77] etc.^{60}

∗ ∗ ∗ ∗ ∗

The algebraic theory of RQA_{α}’s for α ≥ ω is practically the same as that of RQA_{ω}’s, so we will concentrate on the latter. Their algebraic theory is not as extensively developed as that of RCA_{ω}’s, but see Halmos [Ha62], Daigneault–Monk [DM63], [AGN77], Sain [S89], Sain–Thompson [ST89], Pinter [P73], the end of Pinter [P73a]; R. J. Thompson also has results in this line. The next theorem is a generalization of a result of J. S. Johnson, namely of Theorem 10 above.

59BRA, RRA, RCA_{α}, RPA_{n} were all defined by applying SP to algebras of the form ⟨𝔓(^{α}U), ⋯⟩.

60The reason for the popularity of this more complicated similarity type containing all the S_{τ} ’s and C_{(Γ)}’s might have been that the simple schema Σ_{0} quoted above from [ST89] implying all the distinguished axioms like S_{τ∘σ} = S_{τ} ∘ S_{σ} was not available. (Another reason might be that when Halmos [Ha54] introduced quasi–polyadic algebras, awareness of universal algebra as a desirable unifying framework for all branches of abstract algebra in the sense mentioned in §2 and Remark 1 herein, was almost nonexistent. We note that the same problem seems to keep popularity below the deserved level in the case of the unbelievably rich treasure Jónsson–Tarski [JT51] of knowledge on BAO’s.)

THEOREM 13 (Sain–Thompson [ST89]). RQA_{ω} is not axiomatizable by any finite schema^{61} of equations in the sense of [HMTII].

As in the case of RCA_{α}, RRA, and BRA, it is an important open problem to find simple, mathematically transparent, decidable sets Σ of equations axiomatizing RQA_{α} (α > 2).

We could define the RQA_{ω}–theoretic versions of the class L and algebras ℜ𝔣(U) as the algebraic counterparts of the models of first–order logic L_{ωω} without equality as in the RCA_{ω} case. Denoting these by Lp and ℜ𝔣𝔭(U) (where p stands for \underline{p}olyadic), we have that RQA_{ω} is the variety generated by Lp = S{ℜ𝔣𝔭(U) : U is a set}; see^{62} [AGN77, §3.3 Coroll.3.18, p. 28].

We conjecture that the lattice of subvarieties of RQA_{ω} will turn out to be considerably simpler than that of RCA_{ω} described to some extent in Németi [N87, §1.1, p. 245]. It appears to us that developing the theory of RQA_{ω}’s would be quite an important and rewarding task.

In \underline{algebraizing quantifier logics}, the rôle of RQA_{α} is entirely analogous to that of RCA_{α}. Namely, RQA_{ω} is the true algebraic counterpart of classical first–order logic without equality. The logical counterparts of s_{j}^{i} or p_{ij} send the formula φ to φ(v_{i}/v_{j}) or φ(v_{i}/v_{j}, v_{j}/v_{i}) obtained by replacing v_{i} by v_{j} or interchanging v_{i} and v_{j} as free variables of φ respectively. Perhaps the connection with logic on this syntactical level is not as elegant as in the cylindric case, because the operations φ ⟼ φ(v_{i}/v_{j}) are not logical connectives like “∃v_{i}” and “v_{i} = v_{j}” were in the cylindric case. One possible way of avoiding this problem would be to introduce s_{j}^{i} and p_{ij} as logical connectives, restricting atomic formulas to R(v_{0}, v_{1}, …, v_{n}), and treating R(v_{1}v_{0}) as a convenient abbreviation of p_{01}R(v_{0}v_{1}). Of course, this yields an equivalent formulation of L_{ωω} without equality. The connections between RQA_{ω}–theory and logic without equality, including Lp and ℜ𝔣𝔭(U) as algebraic counterpart of models, can be worked out analogously to §4.3 of [HMTII] keeping in mind §5.6 therein, [AGN77], Monk [M71], and Johnson [J73]. At least this is what we conjecture, but this work has not been done yet, at least not in the detail of [HMTII, §4.3], and Németi [N89]. As in the case of cylindric algebras outlined in the remarks after Theorem 4, RPA_{n} (=RQA_{n}) is the algebraic counterpart of the version of L_{n} without equality; see Monk [M71], [J73]. These algebraic counterparts proved rather useful in studying both versions of L_{n}, cf. e.g. the relevant parts in [TG87].

Advanced topics

6. ALGEBRAS FOR LOGICS EXTENDING L_{ωω} 43

If we expand RQA_{α} with the constants Id_{ij} representing equality then we obtain RQA_{α}’s with equality, RQEA_{α}’s, exactly as RPEA_{n}’s were obtained from RPA_{n}’s. Actually, RPEA_{n} = RQEA_{n} for n ∈ ω. Similar observations are true for these algebras, in particular for RQEA_{ω}’s, as were spelled out for the case of α = n(< ω) at the beginning of the present section (§5). But in contrast to Theorem 12, we have to define RQEA_{α} as an SP–closure of set–algebras even if α ≥ ω. One can define the analogs of Crs_{α}’s for RQEA_{α} and for RQA_{α}. The simple finite schema Σ_{0} in Sain– Thompson [ST89] has an RQEA_{α}–theoretic version Σ_{1}, which is also found in [ST89]; Σ_{1} is also finite, and just as simple as Σ_{0}. So, QPEA_{α} = Mod(Σ_{1}) is the finitary variety of quasi–polyadic equality algebras approximating RQEA_{α}. For n < ω, QPEA_{n} = PEA_{n}. An important theorem of Diane Resek implies then that every QPEA_{α} is isomorphic to one of the above mentioned RQEA_{α}– like analogs of Crs_{α}. We do not know whether a similar result holds for RQA_{α}–like analogs of Crs_{α}’s. We note that Andréka generalized Theorem 11 above to RQEA_{α} and RQA_{α} for arbitrary α. By the proof of Theorem 5 one can easily see that RQEA_{α} is a variety. (If not specified otherwise, we always mean “for every ordinal α”). Andréka proved that RQEA_{α} is not axiomatizable over RCA_{α} with any set Σ of universally quantified formulas if Σ contains only finitely many variables and α > 2. For lack of space we do not go into more detail in connection with the theory of RQEA_{α}, but we note that they are basically cylindric algebras enriched with the p_{ij}’s so the two theories should be very close, and the differences illuminating for both.

61A set of individual schemata like {c_{i}c_{j}x = c_{j}c_{i}x, c_{i}Id_{ij} = 1} is called again a schema for simplicity. If the set is infinite then we say that this schema is infinite. Though here we use the definition of a schema in [HMTII] without recalling it, we note that it is the natural generalization of the schema (E0–7) in the present Appendix (§8).

62For some reason, RQA_{α}’s were called representable substitution algebras (and denoted as Rs_{α}) in [AGN77] and its 1974 version. Our present Lp was denoted by Ls_{ω} there.

The joint reducts ⟨𝔓(^{α}U), s_{j}^{i}⟩_{i,j<α} and 𝔓𝔰𝔠_{α}(U) = ⟨𝔓(^{α}U), c_{i}, s_{j}^{i}⟩_{i,j<α} of RQA_{α}’s and RCA_{α}’s have been investigated in Pinter [P73a], Preller [Pr70]. Here we should note that the subreducts RSC_{α} = S {𝔓𝔰𝔠_{α}(U) : U is a set} are already sufficient for carrying through the algebraization of L_{ωω} without equality in a parallel fashion as described above for the RQA_{ω}–case (Pinter [P73b]). Therefore this subreduct RSC_{ω} of RQA_{ω} is not of a completely negligible importance. An axiomatization, analogous to Σ_{0} quoted above from [ST89], of the finitary variety SC_{α} approximating^{63} RSC_{α} can be obtained on the basis of Thompson [T87]. Pinter [P73b] calls the members of SC_{α} substitution–cylindric algebras. The proof of Theorem 12 also yields that RSC_{α} is a variety for α ≥ ω. Sain proved that RSC_{α} is not axiomatizable by any finite schema in the sense of [HMTII] if α > 2. Her proof can be recovered from the proofs of Thm.2(i) in [ST91].

We note that RSC_{ω} = (SRdRCA_{ω}) ∩ (SRdRQA_{ω}) seems to be the weakest reasonable subreduct of RCA_{ω}’s still completely suitable for the algebraization of full first–order logic (without equality). An algebraization of logic completely parallel to the one in [HMTII, §4.3] can be worked out in the same detail for RSC_{ω} in place of RCA_{ω}, with RSC_{ω}’s of universes Rf(U) corresponding to models etc.

\underline{§6. Algebras for logics extending} L_{ωω} ^{64}

Most of the logics we have in mind are discussed in Barwise–Feferman [BF85]. Many of these new logics remain in the realm of finitary relations, for example the finitary logics with additional exotic quantifiers (like “there are uncountably many”, or topological quantifiers of Georgescu [G82a] etc.). These logics are generally denoted by L_{ωω}(Q), or just L(Q), where Q refers to the extra (or exotic) \underline{q}uantifier we are adding to L_{ωω}. Some of these logics are summarized in Part B of Barwise– Feferman [BF85] (but some are scattered all over the book, see e.g. pp.9, 509.)

Many of these quantifiers are given by fixing a function F which associates with every set U a collection F (U) ⊆ P(U) of its subsets. Then Qxφ(x) holds in a model 𝔐 if {a ∈ M : φ(a) is true in 𝔐} ∈ F (M). A special example is the classical quantifier “There exist infinitely many elements x such that φ(x)” suggested by Mostowski in 1957. Other examples are “there exist |U|–many (where U is the universe of the model in question)”, and the standard Qα quantifiers saying “there are ℵ_{α}–many”. Note that when fixing such a logic L(Q), we define the function F for all sets U simultaneously. To make L(Q) satisfy the axioms of abstract model theory, one of which corresponds to permutation invariance in Remark 2 way above, we should place some restrictions on the “universal” function F, but that is not our main concern in this paper.

Let us recall that the universe Rf(U) consists of elements of the form R × ^{ω}U with R ⊆ ^{n}U, for some n ∈ ω. Rf(U) was the universe of the generic example ℜ𝔣(U) of cylindric algebras, i.e. RCA_{ω}’s as well as that of RQA_{ω}’s since Rf(U) is closed under p_{ij} for i, j ∈ ω. Rf(U) consists of all the (dummy embeddings or dummy representations of) finitary relations on U, and as our logic remains finitary, Rf(U) remains adequate for the universe of the generic algebras of relations.

Let us fix F as above. Then the new operator Q : P(^{ω}U) ⟶ P(^{ω}U) is defined as follows:

Q(x) =^{def} {⟨b_{0}, b_{1}, …, b_{n}, …⟩ ∈ ^{ω}U : {u ∈ U : ⟨u, b_{1}, …, b_{n}, …⟩ ∈ x} ∈ F (U)} .

The full Cs_{F} over U is

⟨𝔓(^{ω}U), Q, c_{i}, Id_{ij}⟩_{i,j<ω}

63This is the same kind of approximation as CA_{α} approximates RCA_{α}, cf. the subsection in the middle of §4 and cf. §8. We note that SC_{α} = HSRdQPA_{α} ⫌ RSC_{α}.

64Some algebraizations of some of these logics were already discussed between Theorem 7 and Fact 2 in §4. (These were applications of relativized versions, Crs_{α}’s, of RCA_{α}’s.)

i.e. it is the expansion of the full Cs_{ω} with Q. The representable cylindric algebras with exotic quantifiers determined by F (RCA_{F} ’s) are the algebras embeddable into products of full Cs_{F} ’s. The generic example: ℜ𝔣_{F} (U) is the subalgebra of the full Cs_{F} with universe Rf(U).

The new operation Q : Rf(U) ⟶ Rf(U) acting on our (embedded or “coded”) finitary relations is (for most choices of F) not term definable in RCA_{ω}, in particular, is not such in ℜ𝔣(U), but does not lead out from Rf(U). It produces (codes of) finitary relations from (codes of) finitary relations. We could have introduced Q_{i} for each i ∈ ω by letting Q_{i} = p_{0i} ∘Q∘p_{0i}, but this Q_{i} is expressible in ℜ𝔣_{F} (U) as defined above. The generalization of RCA_{F} ’s from ω–ary relations to α–ary relations for any ordinal α is straightforward. Analogously to the just described RCA case, the new operation Q can be added to RQA_{α}’s, and the other algebras with universe P(^{α}U) discussed so far; it can be added even to RRA’s.) Pinter [P75], Georgescu [G82, G82a], Schwartz [Sw80a] are some of the references to algebras of relations of this kind.

We note the following in connection with Remark 2 (finitization). In practically all of the cases (relevant to logic), F (U) is closed under all permutations of U. In the interesting cases, Q : Rf(U) ⟶ Rf(U) is not term definable in ℜ𝔣(U). Thus expanding RCA_{ω} or RCA_{n} or even RRA with Q is a new possibility in the search outlined in Remark 2, especially because the negative or limiting result of Biró [Bi89] mentioned in Remark 2 above does not apply to Q. However, this observation is mainly of didactical interest, namely a result of Andréka seems to imply that adding such a Q will not result in a finitely axiomatizable variety.^{65}

Concerning the other kinds of quantifiers in Part B of Barwise–Feferman [BF85], like Henkin quantifiers, we do not know of any work on their algebraization. We feel that one should be able to stay inside the universe Rf(U) when algebraizing them, too.

Algebras for infinitary logics

Let α be an arbitrary ordinal. After having introduced and studied RQA_{α}’s in [Ha54], Halmos expanded them to what are now called representable polyadic algebras of α–ary relations (RPA_{α}’s), cf. [HMTII, §5.4], Halmos [Ha62]. The new operations are the natural infinitary generalizations of the s_{j}^{i}, p_{ij}, and c_{i}’s of RQA_{α}’s. Next we generalize the operation C_{(Γ)} from finite Γ ⊆ α to arbitrary Γ ⊆ α. Let x ⊆ ^{α}U and Γ ⊆ α. Then

C_{Γ}(x) =^{def} C_{(Γ)}(x) =^{def} {q ∈ ^{α}U : (∃s ∈ x)q ↾ (α ∖ Γ) = s ↾ (α ∖ Γ)}.

Note that c_{i}x = C_{{i}}x and C_{{i,j}}x = c_{i}c_{j}x = c_{j}c_{i}x, and the same for any finite set {i_{1}, …, i_{n}} ⊆ α. For arbitrary τ ∈ ^{α}α, the operation S_{τ} has already been defined in §4.

By an RPA_{α} we understand an algebra embeddable into a direct product of algebras of the form ⟨𝔓(^{α}U), C_{Γ}, S_{τ} : Γ ⊆ α and τ ∈ ^{α}α⟩.

Note that, though for α = n < ω we have two different definitions of RPA_{n}, they are equivalent (up to term definitional equivalence), i.e. they differ only in the choice of the fundamental operators (like BA’s of the form ⟨B, ∧, ∨, −, 0⟩, or ⟨B, ∧, ∨, −, 1⟩, or ⟨B, ∧, −⟩).

For α < ω we already discussed these algebras. So let α ≥ ω for a while.

65Andréka proved that if Q defined by F is permutation invariant (for all p : U ↣ → U, p(F (U)) = F (U)), then adding one or more of these Q’s (each may be based on a different but invariant F) does not make RCA_{n} finitely axiomatizable or RCA_{ω} finite schema axiomatizable, for n > 2. Moreover, she proved the same for any new permutation invariant unary operator f : P(^{n}U) ⟶ P(^{n}U) satisfying the equation f(x ∪ y) = f(x) ∪ f(y)

in place of Q. So adding such a new operation f to RCA_{n} does not make RCA_{n} finitely axiomatizable. Thus Problem 2.2 cannot be solved by adding such f’s or Q’s (to the old operations). She also proved that the weaker version (of the problem) quoted from [TG78] at the end of Remark 2 is not solvable either by adding such f’s or Q’s. I.e. she proved that no finitely axiomatizable variety K_{0} containing the version K^{+} of RCA_{n} (n > 2) expanded by such f’s or Q’s has the property RCA_{n} = SRdK_{0}. (Sain proved that this last negative result does not extend to RCA_{ω} from RCA_{n}. Cf. [S87, 87a].) These negative results of Andréka generalize to polyadic algebras with equality, and remain true if we add the operation trc of taking transitive closure to RCA_{n}. Here, if R ⊆ U × U then trc(R ×^{n−2}U) = (R^{∗} ×^{n−2}U) where R^{∗} is the usual transitive closure of the binary relation R. (These were proved by Andréka.) (The case of adding binary invariant operators remains wide open.) Cf. e.g. [A91, Thm.4 and Remark 6].

THEOREM 14. RPA_{α} is a discriminator variety in the standard universal algebraic sense (see [BS81], [AJN88]). So are BRA, RRA, RCA_{n}, and RPA_{n} if n < ω.^{66}

We cannot speak about finite or even recursively enumerable or decidable axiomatizations in connection with RPA_{α}’s, because they have at least continuum many basic operators. Intuitively, this large number of operations seems to be a drawback of RPA_{α}’s when one is trying to apply them to finitary logics or to algebras of arbitrary finitary relations. The main bulk of these operations seems to be irrelevant for L_{ωω} or for {R : R ⊆ ^{n}U and n ∈ ω}, and therefore seems to “pull” the theory in misleading directions. On the other hand, for infinitary logics these operations are useful and relevant, cf. e.g. Ex.5.6.3, 5.6.6, 5.6.9 in [HMTII]. Further, RPA_{α}’s are very useful via their reducts as we will soon see.

One could argue that postulates like {S_{τ∘σ}(x) = (S_{τ} ∘ S_{σ})(x) : τ, σ ∈ ^{ω}ω} could be considered as some kind of generalized finite schemata because they look “schematic” in some sense. However, the intuitive value (or concreteness) of these generalized schemata is more problematic than that of the c_{i}c_{j} = c_{j}c_{i} kind because if e.g. we are given three recursive functions τ, σ, ρ ∈ ^{ω}ω by their recursive definitions then it might be an unsolvable problem to decide whether S_{ρ}(x) = S_{τ} S_{σ}(x) is an instance of the above schema^{67}. In connection with Theorems 13, 10, 4 we note that Daigneault and Keisler independently proved the existence of a finite generalized schema of equations in the above sense axiomatizing the variety RPA_{α} (recall that α ≥ ω), cf. [HMTII, items 5.4.41 and 5.4.1], Daigneault–Monk [DM63]. Pinter [P73b] simplified this generalized schema considerably. Pinter’s simplified axioms for RPA_{α} proved rather useful in subsequent work (e.g. of R. J. Thompson). Results on the nature and complexity of the presently discussed “polyadic kind” (s_{τ∘σ}x = …) of schemas are in Németi–Sági [NS96].

Returning briefly to the \underline{connection with logic}^{68} as approached in the RCA_{ω} and RQA_{ω} cases, recall that ℜ𝔣𝔭(U), the quasi–polyadic algebra of all finitary relations over U, was introduced below Theorem 13 in an analogous manner to ℜ𝔣(U) introduced below Theorem 5. Let L_{p}^{∞} be the class of subalgebras of RPA_{ω}’s with universes of the form Rf(U), that is the RQA_{ω} algebras ℜ𝔣𝔭(U) of (embedded) finitary relations over U, but now expanded with the operations S_{τ}, C_{(Γ)} for τ ∈ ^{ω}ω and Γ ⊆ ω. Similarly to the previous cases, L_{p}^{∞} consists of the polyadic algebras corresponding to models of L_{ωω}. (Further SPL_{p}^{∞} contains all the algebras corresponding to other aspects, e.g. theories of L_{ωω})

The following is in sharp contrast with what we saw for RCA_{ω}’s and RQA_{ω}’s.

THEOREM 15. RPA_{ω} ≠ HSP(L_{p}^{∞}). That is, the variety generated by L_{p}^{∞} is strictly smaller than RPA_{ω}.

Proof: ^{69} Let τ : ω ↣ → ω be a permutation acting as predecessor on the even numbers, and as a successor on the odds. I.e. τ(0) = 1 and τ(2n + 2) = 2n, τ(2n + 1) = 2n + 3. Let e(x) be the equation

C_{(α)}(x ⊕ S_{τ} x) ≥ C_{(α)}(x ⊕ C_{(α)}x)

66Discriminator varieties are getting more and more into the focus of attention in algebraic logic, see e.g. [AJN88], Blok–Pigozzi [BP89a], [BP89b], Pigozzi [P89]. (Every discriminator variety with two term definable constants is very close to being a variety of BAO’s, see the last two references, and [BP89b].) As mentioned in the footnote of Remark 1 in §3, our classes are more than discriminator, they are doubly pointed discriminator, and hence stronger results are available.

67Indeed, Gábor Sági proved the following. Let Ax be the set of PA_{ω}-axioms given by a finite schema in [HMTII].

There are total recursive functions τ_{0}, …, τ_{3} such that if L is the sublanguage of the equational language of PA_{ω} involving only τ_{0}, …, τ_{3} as substitutions and no cylindrifications, then the following holds. The set {e ∈ L : Ax ⊢ e} is not recursively enumerable. This seems to point in the direction that the polyadic axiom schemas might be less constructive or less descriptive than the cylindric-like ones. Cf. Sági [Sa95], [NS96].

68We will return to this connection again with a greater emphasis on infinitary logics.

69The idea of this proof occurs in the proof of Thm.9(i) in the June 1987 version of Sain [S87a], and in Sain [S89].

where ⊕ denotes symmetric difference. Then L_{p}^{∞} ⊧ e(x), but RPA_{ω} ⊭ e(x) because e(x) fails for e.g. x = {⟨u, u, …, u, …⟩ : u ∈ U}, as well as for x = {q ∈ ^{ω}U : {i : q_{i} ≠ b} is finite}, for any fixed b ∈ U if |U| > 1.

The above theorem points in the direction of the limited relevance of RPA_{ω}–theory for finitary quantifier logics. Namely, there are polyadic equations valid in all algebraic counterparts of finitary logics which are not valid in RPA_{ω}, hence investigating RPA_{ω} in general might lead one to questions irrelevant for finitary logics^{70}. (The above equation distinguishes the whole of RPA_{ω} from polyadic algebraic counterparts of aspects or “parts” other than just models of finitary logics like the Lindenbaum–Tarski algebras of formulas, of those of theories etc.) Of course, this does not diminish the value of RPA_{ω} for investigating infinitary logics.

There is a way, however, in which RPA_{ω}’s seem to be very useful for all parts of algebraic logic, namely they provide us with a wealth of subreducts to study. Indeed, for any subsemigroup ⟨S, ∘⟩ ⊆ ⟨^{ω}ω, ∘⟩ of the full transformation semigroup ^{ω}ω of ω we have a potentially interesting “reduct” generated as a variety by the algebras of the form ⟨𝔓(^{ω}U), c_{i}, S_{τ} : τ ∈ S, i ∈ ω⟩. RQA_{ω}’s are the case when S_{1} = “all finite transformations”, and Pinter’s substitution–cylindric algebras obtained from ⟨𝔓(^{ω}U), c_{i}, s_{j}^{i}⟩_{i,j<α} are the case when S_{2} = “subsemigroup of S_{1} generated by {[i/j] : i, j ∈ ω}”. Let RPA_{S} be the variety associated above to the semigroup S. For many interesting choices of S with an arbitrary α ≥ ω in place of ω, there are deep results in Daigneault–Monk [DM63] which seems to be one of the key sources for results on polyadic algebras and their generalizations like RPA_{S}.

THEOREM 16 (Sain [S87a, Thm.0, Corollary 17]. There is a subsemigroup S of ^{ω}ω such that

(i) RPA_{S} is term definitionally (i.e. polynomially) equivalent with a variety V_{S} \underline{axiomatizable} \underline{by finitely many} equations.

(ii) RQA_{ω} = SRd(RPA_{S}) i.e. the class of representable quasi–polyadic algebras coincides with the appropriate subreducts of the class RPA_{S}.

The above result is a positive solution for the finitizability problem which we have discussed throughout this paper beginning with Remark 2. In particular, Theorem 16 solves positively the RQA_{ω} theoretic version of Problem 4.1 in §4. As we indicated in Remark 2, this problem is really a whole family of problems (or a theory built around problems driven by a fundamental desire for insights in connection with a mysterious pattern of recurring negative results which might be the symptoms of some uncovered property of first–order logics). Therefore a single positive result, even if as powerful as the above one, cannot be the final word about the finitization problem, rather the opposite is the case: it is a new beginning in finitizability theory. In our opinion, the above result is a breakthrough. Namely, all elements of RPA_{S} are representable as (i.e. are isomorphic to) algebras of ω–ary relations and all operations of RPA_{S}’s are permutation invariant set theoretic ones (obviously, since they are S_{τ} ’s and c_{i}’s); and at the same time, RPA_{S} is a finitely axiomatizable variety if we choose its basic operations properly. The latter of course implies that the clone (set of term functions) of RPA_{S} is finitely generated (as we mentioned at discussing the finitization problem of RCA_{ω}). Making the clone of RPA_{S} finitely generated is only the first step of the proof, the easiest one. Namely, it is easy to expand RCA_{ω}’s, or RQA_{ω}’s with S_{τ} ’s such that the clone of the expanded class becomes finitely generated, see Copeland algebras in [HMTII, p. 264]. However, as indicated therein, it is very hard to force the new algebras to become finitely axiomatizable. Even if the original class was finite schema axiomatizable (which is not the case with RCA_{ω} or RQA_{ω}), this property tends to go away as a by–product of our coding the original infinity of operations by finitely many new ones, cf. Demaree [D72, Corollary 3.6, p. 654]. The hardest part, however, seems to be forcing the new operations to be representable, too (by adding the new operations, one forces the old ones to be representable; but then the problem is, roughly, that nothing forces the new ones to be representable).

70Since the opposite of Theorem 15 is true for RCA_{ω} and RQA_{ω}, this danger is not present there.

In §4 (Application to logic) of Sain [S87a], she shows that RPA_{S} is indeed adequate for the algebraization for first–order logic without equality. Actually [S87a] works out the connections with logic in a way parallel with [HMTII, §4.3].

There are important open problems stated in Sain [S87a]; one of them is to simplify the finite set of equations describing RPA_{S} and another one is to simplify the class (or operations of) RPA_{S} itself. Actually [S87a] presents some very nice, simple and elegant candidates S to take the place of S, the only problem left open is to decide whether one of these S’s is finitely presented as a semigroup. A positive answer would provide us with an intuitively illuminating, transparent and very attractive version of RPA_{S} in Theorem 16 above. In this connection we should mention that R. J. Thompson has a result which is also highly relevant to the finitization problem. His class RPA_{T} is also a finitely axiomatizable subreduct of RPA_{ω}; but RQA_{ω} or even the subreduct RSC_{ω} of RQA_{ω} containing only the s_{j}^{i}’s and c_{i}’s (or even the one containig the c_{i}’s only) is not obtainable as a subreduct of RPA_{T}. So adequateness of RPA_{T} for algebraizing first–order logic seems to be problematic. (On the other hand, the RPA_{T} operations when applied to Rf(U) generate the same subalgebras as those of ℜ𝔣𝔭(U).)

The above important results, in a sense, form a part of a relatively recent movement in algebraic logic connected with polyadic algebras. This movement looks at classes K of subreducts of RPA_{ω}’s such that K contains all RQA_{ω}’s as subreducts. Actually Pinter’s simpler substitution algebras RSC_{ω} = SRd(RQA_{ω}) introduced at the end of §5 above are sufficient instead of RQA_{ω}’s. For simplicity, we look at

(5.1)  RQA_{ω} = SRdK  and  K = SRd(RPA_{ω}) .

Theorem 16 above shows that the equational theory of such a class K can be finitely axiomatizable. According to Thm’s 7(i), 9(i) on pp. 14–15 (proved on p. 15) of the June 87 version of Sain [S87a], there are choices of K satisfying (5.1) such that K is only a quasivariety but not a variety. (It is also proved there that HK is axiomatizable by a finite schema of equations for the same K. It is an interesting open problem whether that K is axiomatizable by a finite schema of quasi–equations.) This means that when looking for simpler solutions for the finitization problem than the one in Theorem 16, then it might be a good idea to look for finitely axiomatizable quasivarieties too, and not only for varieties. The works of Blok and Pigozzi (cf. [BP89]) indicate that quasivarieties are just as good vehicles for algebraic logic as varieties. Works concerning classes like K above are numerous, but we should point out Craig’s semigroup oriented school (cf. earlier references to Craig’s works, Howard [H65], Thompson’s works), Demaree [D72], Daigneault–Monk [DM63] — without claiming that we have mentioned all the important ones.

THEOREM 17 (Sain-Gyuris [SGy94]Thm.2.5). There is a subsemigroup S of ^{ω}ω such that

(i) S{⟨𝔓(^{ω}U), c_{i}, s_{τ} : τ ∈ S, i ∈ ω⟩ : U is a set} = ModQax for some finite schema Qax of quasi-equations; and

(ii) RQA_{ω} = SRd(ModQax).

Let RPA_{S}^{+} be the class of Theorem 17 (i).

THEOREM 18 ([SGy94]Thm.2.5). Let S be as in Thm.17. Then RPA_{S}^{+} expanded with c_{(ω)} forms a finite schema axiomatizable variety.

Theorems 17,18 improve on Thm.16 the following ways:

(1) The axiom systems involved (e.g. Qax) are quite simple and transparent. (2) The class RPA_{S}^{+} is tighter than RPA_{S} in that the greatest element of any RPA_{S}^{+} is a Cartesian space. The price we pay for these improvements is that in Theorems 17,18 we have only a finite schema of axioms while in Thm.16 we have a truly finite set of axioms. It remains an \underline{open problem} to improve Thm.16 in the style of Thm.17, i.e. to finitely axiomatize the quasi-variety RPA_{S}^{+} (⊆ RPA_{S}) with the choice of S as in Thm.16.

The positive results Theorems 16,17 have been extended to algebras RQPEA_{S} of logic with equality, too in [SGy94], [S92]. However, there compactness of the logic had to be sacrificed as follows.

Let RQPEA_{S} be the expansion of RQPA_{S} with the constants Id_{ij} representing equality (v_{i} = v_{j}). Then RQPEA_{S} is not closed under ultraproducts for any S which contains at least one infinitary substitution τ ∈ ^{ω}ω (cf. Sain [S96]). Therefore, the only thing we can try is to axiomatize Eq(RQPEA_{S}) but not RQPEA_{S} itself. Indeed, finite (and finite schema) axiomatizations of Eq(RQPEA_{S}) have been achieved in Sain [S92] and Sain-Gyuris [SGy94] for various choices of S.

Eq(RPEA_{ω}) was investigated in Németi-Sági [NS96]. E.g. Eq(RPEA_{ω}) is not axiomatizable by any finite or even recursively axiomatizable schema.

In passing we note that expanding RCA_{α}’s with the single operations C_{(α)} yields a class RCA_{α}^{+} highly relevant to L_{ωω} and having a smoother theory in some respect than RCA_{α}’s for α ≥ ω. E.g. RCA_{α}^{+} is a discriminator variety, while RCA_{α} is not if α ≥ ω. (The logical meaning of C_{(α)} is taking the universal closure of a formula.)

Let us return to \underline{algebraizing infinitary logics}. For the rôle of RPA_{α}’s (α ≥ ω) and their reducts and expansions already mentioned in this paper, see [HMTII, Examples 5.6.2, 5.6.5, 5.6.8], Keisler [K63], Daigneault–Monk [DM63].^{71} Other but not unrelated algebraizations of infinitary logics are in Preller [Pr68a, Pr69, Pr69a], Lucas [L68].

Lucas [L68] is related to observing that if we want to treat infinitary logic with equality via RPA_{α}’s then it is not enough to add the Id_{ij}’s as new constants, but we will also need constants denoting the intersections of infinitely many of the Id_{ij}’s.

The transformational algebras in Craig [Cr74a, p. 30] achieve this effect in a very elegant way, but the approach there is more ambitious than aiming for just this. Namely, one of the purposes of that work is unification of (a very large portion of) algebraic logic.^{72} For any τ, besides S_{τ}, Craig introduces T_{τ} : P(^{α}U) ⟶ P(^{α}U) as follows. T_{τ} (x) = {τ ∘ q : q ∈ x}, for any x ⊆ αU. We note that T_{τ} is a conjugate of S_{τ} in the sense of Jónsson–Tarski [JT51], and that the term function t_{j}^{i}(x) = Id_{ij} ∧ c_{i}x of CA_{α}’s is the same as T_{[i/j]}. Let S be a subsemigroup of the transformation semigroup ^{α}α. Then the class TB_{S} of S–transformational algebras consists of the algebras embeddable into products of algebras of the form ⟨𝔓(^{α}U), S_{τ} , T_{τ} ⟩_{τ∈S}. Observe that c_{i}x = S_{[i/j]}T_{[i/j]}(x) if j ≠ i, moreover, to each Γ ≠ α there is τ ∈ ^{α}α such that C_{(Γ)}x = S_{τ} T_{τ} (x). But then, C_{(α)}(x) = C_{({0})}C_{(α∖{0})}(x) is expressible, too. So the TB_{S}’s are more powerful than the S–polyadic algebras^{73}, i.e. all S–polyadic operations are expressible, moreover Id_{ij} = T_{[i/j]}(1) shows that even more are expressible in TB_{S}. If S is generated by a subset G then, of course, it is enough to consider ⟨𝔓(^{α}U), S_{τ} , T_{τ} ⟩_{τ∈G}. If G = {[i/j] : i, j < α}, then TB_{S} is the same as RCA_{α}. If S consists of the finite transformations of α then TB_{S} is RQEA_{α} i.e. quasi–polyadic algebras with equality. If S = ^{α}α then TB_{S} coincides with Lucas’s expanded RPEA_{α}’s. (The latter are basically RPA_{α}’s with new constants Id_{E} for equivalence relations E on α.) There are further examples in Craig [Cr74a], we listed the above ones to illustrate the unifying power and elegance of this approach.

71A problem in connection with algebraizing infinitary logic is the following. As mentioned in the discussion of RCA_{ω}’s around the end of §4, the algebraic counterparts of models of L_{ωω} form the class L which admits an intrinsic characterization condensed into the notation L = Cs_{ω}^{reg} ∩ Lf_{ω} in [HMTAN] and [HMTII]. In infinitary L_{κλ}, it is customary to have at least α = |κ + λ|^{+} variables. This leads to the reducts of RPA_{α}’s in Daigneault–Monk [DM63] in which C_{(Γ)} is restricted to the case when |Γ| < α. But then the class of algebraic counterparts of models (analogous to L) is not the polyadic version of Cs_{α}, nor the intersection of that of Cs_{α}^{reg} with any abstract class (generalizing Lf) as was shown in [N78] (based on Andréka [A73]). So a sleek (like Cs^{reg}) characterization of the algebraic counterparts of models in the infinitary case seems to need a new idea. Some nonconclusive experiments in this direction were reported in [N78].

72Different approaches for a similar kind of unification are in Schein [Sc70], Andréka–Németi [AN80], cf. [HMTII, p. 260] and Andréka [A77], Salibra [Sa89], and on a different, more universal algebraic level of abstraction [HMTII, pp.255–260], cf. Andréka–Sain [AS78] and [ANS84].

73These are basically the RPA_{S}’s, but sometimes C_{Γ} is also included for those choices of Γ for which |τ(Γ)| = 1 for some τ ∈ S.

In passing we note that if K is a reduct of TB_{S} (S = ^{α}α) containing C_{(α)} and the Boolean operations as derived operations, then HSPK is a discriminator variety, cf. Theorem 14.

§7. Other algebras of relations

We have no reason to believe that the decisions we made (at certain “branching points”) in the development outlined so far were the best possible ones. (One aspect of this observation already appeared as a sequence of recurring remarks starting with Remark 2 addressing the finitization problem.) Many of the approaches listed below originate from making some of these decisions differently.

As mentioned in the introduction, this section was intended to be somewhat complementary to a similar overview at the end of [HMTII]. Therefore, in this section we tend to restrict ourselves to giving such references which were not listed in [HMTII] (e.g. because they appeared later than [HMTII]).

(1) Many–sorted cylindric algebras

In the middle of §4, our task was to glue the algebras

ℜ𝔢𝔩_{2}(U) = ⟨𝔓(^{2}U) …⟩, …, ℜ𝔢𝔩_{n}(U) = ⟨𝔓(^{n}U) …⟩, … n ∈ ω

belonging to the varieties RCA_{2}, …, RCA_{n}, … respectively, together into a single algebra of relations of arbitrary finite ranks. This task admits an easy solution if we are willing to use many–sorted (in other words heterogeneous) algebras (cf. Burmeister [Bu86], Ehrig–Mahr [EM85], Lugowski [L76] for the many–sorted version of universal algebra). Many–sorted cylindric algebras, MsCA’s have the ℜ𝔢𝔩_{n}(U)’s as sorts for each n < ω. So, in particular, there are ω–many sorts, and the old operations of ℜ𝔢𝔩_{n}(U) act on the sort or universe P(^{n}U). We need operations connecting the different sorts too, so let c_{n}^{+} : P(^{n}U) ⟶ P(^{n+1}U) be our old dummy embedding, c_{n}^{+} : R ↦ R×U for R ⊆ ^{n}U. Its “inverse” c_{n}^{−} : P(^{n+1}U) ⟶ P(^{n}U) deletes the last column of any n + 1–ary relation, e.g. c_{2}^{−}{⟨a, b, c⟩} = {⟨a, b⟩}. So in particular c_{n}^{+}c_{n}^{−}(x) = c_{n}(x) for x ⊆ ^{n+1}U. Now, a full MsCA is of the form

(7.1)  ⟨ℜ𝔢𝔩_{n}(U), c_{n}^{+}, c_{n}^{−} : n < ω⟩ .

Then one defines MsCA’s by taking subalgebras of direct products. By using results from cylindric algebra theory (cf. [HMTII, 2.3.8, p. 63]), one can rearrange these algebras such that they form a many–sorted variety axiomatizable by a finite schema of equations, see [HMTII, p. 263], Bernays [B59]. Other works on this MsCA approach are e.g. in Börner [B88], Schwartz [Sw79]. In passing we note that this many–sorted approach seems to be connected with the universal algebraic theory of clones (as was made explicit in Börner [B88]). Namely, clones are sets of special finitary relations whose speciality is that they are functions. Therefore an algebra of clones over a set U is at the same time an algebra of (certain) finitary relations over U. The many–sorted algebraic approach to clones is one of the oldest, most traditional ones, cf. Cohn [C65, §III.3 and Exercise 3], for example.

Despite of the fact that this many–sorted approach leads to a natural positive solution of the schema version of the finitization problem (cf. Remark 2 in §3, Problem 4.1 at the end of §4 etc. for this problem), as far as we know, it hasn’t been explored too much beyond proving (incarnations of) this result.^{74} We do not know the reason for this.

74Recall (from §4) that the schema version is the less ambitious version of the finitization projects.

(2) Partial cylindric algebras

We mentioned clones above as motivating examples for the many–sorted approach. Clones are also often treated as partial algebras (cf. McKenzie et al. [MMT, p. 143], Cohn [C65, Exercise 3 in §III.3]). For the theory of partial algebras see Craig [Cr89], Burmeister [Bu86], Andréka– Németi [AN76]. Actually, many–sorted algebras are well known to be special cases of one–sorted partial algebras. Therefore an alternative way of doing (1) above is using partial algebras. (Boole originally introduced his algebras as partial ones, anyway.)

Let Frl(U) = {R : R ⊆ ^{n}U for some n < ω} be the collection of finitary relations over U (as above Fact 2 in §4). If ∅ ≠ R ⊆ ^{n}U then −R = ^{n}U ∖ R, otherwise − is undefined. R ∪ S and R ∩ S are defined only if the ranks of R and S coincide. Let R ⊆ ^{n}U. For i < n, c_{i}(R) is the usual, for i ≥ n we let c_{i}(R) = R. c_{i}^{+} and c_{i}^{−} are as in subsection (1) above, and are defined only on R ⊆ ^{n}U and R ⊆ ^{n+1}U respectively. For each n < ω, we add the constant Id_{n} = (Id ↾ (^{n}U)) = {q ∈ ^{n}U : q_{0} = q_{i} for all i < n}. Now our generic algebras are of the form

⟨Frl(U), ∩, ∪, −, c_{n}, c_{n}^{+}, c_{n}^{−}, Id_{n}⟩_{n<ω} .

PaCA’s are the algebras embeddable into direct products of these (as usual). Since nU = cc_{0} … c_{n−1}(Id_{n}) is a derived constant, each sort ℜ𝔢𝔩_{n}(U) of the algebras in (1) above is recoverable in the present partial algebraic setting too.

We note that the point in using the new, partial versions of ∪, ∩, and − is that they do not lead out from the circle of the usual finitary relations, i.e. Frl(U) is closed under them. The main problem with the total versions of these operations was, noted immediately above the beginning of §3, that both total complementation and total union led out of Frl(U), e.g. the union of relations of different ranks is a set of sequences of various length hence it is not a relation in the usual sense.

Despite of the availability of partial algebra theory, almost nothing is known about these algebras, as far as we know.

(3) Algebras of sets of finite sequences

Let us recall the BA 𝔓(^{<ω}U) of finite sequences from before §3. Let succ : n ↦ n + 1, pred : n + 1 ↦ n with pred(0) = 0 be the usual transformations of ω. Then T_{succ}, T_{pred}, S_{succ}, S_{pred} are defined on subsets of ^{<ω}U too, since for any q ∈ ^{<ω}U we have succ ∘ q, pred ∘ q ∈ ^{<ω}U, too. E.g. succ ∘ ⟨a, b, c⟩ = ⟨b, c⟩ and pred ∘ ⟨a, b, c⟩ = ⟨a, a, b, c⟩. For R ⊆ ^{<ω}U, by definition S_{succ}(R) = {q ∈ ^{<ω}U : succ ∘ q ∈ R} and similarly for S_{pred}. The definition of T_{succ}, T_{pred} is even more literally the same as it was around the end of §6 above.

By a finite sequence algebra (FSA) we understand an algebra embeddable into a direct product of algebras of the form

⟨𝔓(^{<ω}U), c_{i}, S_{succ}, S_{pred}, T_{succ}, T_{pred}⟩_{i<ω} ,

where c_{i} is the usual one, e.g. c_{2}{⟨a, b, c⟩} = {⟨a, b, u⟩ : u ∈ U}, c_{3}{⟨a, b, c⟩} = {⟨a, b, c⟩}, and c_{i}(x ∪ y) = (c_{i}x) ∪ (c_{i}y). Recall that 1 denotes the top element ^{<ω}U in BA language. We note that ^{1}U = {⟨u⟩ : u ∈ U} = −c_{0}T_{pred}(1), ^{0}U = {⟨⟩} = T_{succ}(^{1}U), ^{2}U = S_{succ}(^{1}U), …, n+1U = S_{succ}(^{n}U) for n > 0, finally Id_{2} = {⟨u, u⟩ : u ∈ U} = T_{pred}(^{1}U) and Id_{n+1} = T_{pred}(Id_{n}) for n > 0. Since ^{n}U and Id_{n} are derivable constants, and the c_{i}’s and the standard BA operations are built in, the cylindric algebra ℜ𝔢𝔩_{n}(U) is available as the set of elements below ^{n}U.

So the many–sorted cylindric algebra with n–th sort P(^{n}U) or ℜ𝔢𝔩_{n}(U) described in (7.1) in subsection (1) above is contained in a sense in our present FSA with universe P(^{<ω}U). Moreover, all the operations of the many–sorted algebra in (7.1) are definable in our present FSA. Despite of all these nice things, the algebraic theory of FSA cannot replace that of MsCA because e.g. the algebraic operators like direct products or ultraproducts behave rather differently in the two cases. For example, the direct power or ultrapower with exponent ω of the above FSA will contain elements x, y corresponding to the ω–sequences ⟨^{2}U,^{3}U, …,^{n}U, …⟩ and ⟨Id_{2}, Id_{3}, …, Id_{n}, …⟩ respectively.

Elements like x and y do not fit into any of the sorts of MsCA’s, and indeed they do not appear in direct powers or ultrapowers of MsCA’s.

Finally we point out that Craig’s algebras as described in [HMTII, item (3), p. 265] are term definably equivalent with FSA’s above. To see this, we note that, using the notation Q(x) and P(x) introduced therein, we have Q(x) = S_{succ}(x) ∖^{0}U and P(x) = T_{succ}(x ∖^{0}U). This is enough for one direction. In the other direction T_{pred}(x) = (Id_{01} ∩ Q(x)) ∪ (x ∩^{0}U), where ^{0}U = −Q(1), and similarly for the rest of the FSA operations.

For interesting results concerning FSA see op. cit., and the references therein, e.g. Craig [Cr74a, Cr74], Monk [M70].

FIGURE 4

(4) Algebras of generalized finite sequences

One disadvantage of working with ^{<ω}U as above is that these finite sequences impose a structure ⟨ω, succ, <⟩ on the index set. I.e. the indices 1,2 and say 5 are not equivalent as they were in the RCA_{n} or RCA_{ω} or RQA_{ω} case. Another symptom of the same “asymmetry” is that p_{12} = S_{[1,2]} cannot be applied to P(^{<ω}U) because it leads out of P(^{<ω}U). Namely, S_{[1,2]}{⟨a, b⟩} = {[1, 2]∘⟨a, b⟩} but f = [1, 2] ∘ ⟨a, b⟩ is not a finite sequence since f(0) = a, f(1) is undefined, f(2) = b.

Let Gfs(U) = ⋃{^{H}U : H ⊆ ω and |H| < ω}. That is, a generalized finite sequence is a function mapping a finite set of natural numbers into U. Gfs(U) is the set of these. Now, replacing ^{<ω}U with Gfs(U) in subsection (3) above, we obtain the definition of GFSA’s, i.e. generalized FSA’s. The theory becomes smoother this way. Moreover, we may safely add S_{[i/j]}, S_{[i,j]} to the operations. If after adding these we remove the ones associated with pred and succ then we obtain something very close to RQA_{ω}’s or RCA_{ω}’s, depending on whether we keep Id_{2}.

In many respect the latter version of GFSA’s are very close to what are called weak cylindric set algebras in [HMTII]. See e.g. Howard [H65], Craig [Cr65, p.58], [Cr91], [Cr74], Andréka [A77] for GFSA’s. The GFSA approach was first discovered by Howard [H65], cf. Craig [Cr65], [Cr91]. In passing we note that this approach of generalized sequences allows for the fulfillment of an old dream of some algebraic logicians: If we want, we can introduce c_{i}^{−} as an operator of simply deleting i from the arguments of all the sequences, i.e. c_{i}^{−}(x) = {f ↾ (Dom(f) ∖ {i}) : f ∈ x}. Then c_{i}^{−} really deletes (unmistakably abstracts away from) the i–th column of any relation (in algebras of relations coming from computer science this deletion operation is frequent). In general, this GFSA approach seems to be close to the way relations are often approached in computer science.

(5) Making function symbols of logics explicit

In subsection (1) of this section we already noted that the algebra of clones is a well investigated example of special algebras of finitary relations which happen to be functions, see the references in (1). The operations of such algebras of finitary functions will have a different character from those discussed so far, because in choosing them, we have to make sure that they yield functions when applied to functions (so even the Boolean operations are excluded).

To any set U, we can associate two algebras, one, say 𝔉𝔫(U), of finitary functions, and an RCA_{ω}, say ℜ𝔣(U), for finitary relations over U. Since their elements are strongly related, we can combine them into a single two–sorted algebra 𝔗𝔰(U) = ⟨𝔉𝔫(U), ℜ𝔣(U), f_{0}, …, f_{n}⟩, where the f_{i}’s are operations acting between the sorts. Algebras like 𝔗𝔰(U) can be used in algebraizing quantifier logics with function symbols in such a way that terms (and term–like syntactic entities of the language) correspond to 𝔉𝔫(U), and formulas correspond to ℜ𝔣(U). So when forming the Lindenbaum–Tarski algebra of a theory then (the equivalence classes of) terms form one sort of the algebra, and (those of) formulas form another. In general, when a logic has several syntactic categories often called types in computer science logics (see Salibra [Sa89]), then each syntactic category or type gives rise to a separate sort. Such an approach was elaborated in Andréka–Sain [AS78] and subsequent works reviewed in [HMTII, pp. 250–260], see Examples 5.6.3, 5.6.6, 5.6.9 for the treatment of the “term sort” mentioned above. Feldman [Fe89], Cirulis [C91, C88, C88a, C86a], [MP88] represent a variety of approaches, and contain further results in connection with this topic.

(6) Cylindric–like algebras for nonclassical quantifier logics

See e.g. items 4–6, 18 on pp. 265–270 of [HMTII], Freeman [F76], Monk [M60], Georgescu [G79, G79a], Schwartz [Sw80], Ferenczi [F89a], Diskin [Dis90], [Dis91], §4 of [M77], and item (7) below. This is a rather incomplete list. What item (11) below writes about nonclassical (or non–Boolean) generalizations of Boolean Algebras with Operators is relevant here.

(7) The Rasiowa–Sikorski approach

For a detailed exposition of the Rasiowa–Sikorski approach (RS–approach from now on) to algebraizing logic (and in particular to applying algebra in logic) the reader is referred to Rasiowa– Sikorski [RS63] (but see also the works of Rasiowa listed in the References). Here we will restrict ourselves to mentioning some of the relationships between the RS–approach and the approaches reviewed in more detail in the present paper.^{75}

Instead of representing the first–order connectives by adding the infinity of new operations c_{i}, s_{j}^{i} (i, j ∈ ω) (or d_{ij} instead of s_{j}^{i}) to BA’s, we could add one infinitary partial operation ⋂. Then (generic examples of) the new algebras look like ⟨𝔓(U), ⋂⟩ where

⋂(X) = the set theoretic intersection of X ⊆ U whenever ⋂ is defined on X.

In particular, ⋂ : Dom(⋂) ⟶ P(U) with Dom(⋂) ⊆ PP(U).^{76} The theory of partial algebras is quite well developed (cf. Andréka–Németi [AN76], Burmeister [Bu86], Craig [Cr88], [ACrN88], works of Bartol et al.), and has been extended to infinitary operations, so partiality is no obstacle in algebraizing logics this way. These new partial algebras (BA’s + ⋂) are called generalized BA’s [[RS63] pp.27–30] or Q–BA’s ([RS63, p.42, p.68]), where “Q” refers to the domain of ⋂ which need not contain all subsets of the Boolean algebra in consideration. The letter “Q” also refers to the fact that the new infinitary operation is used for algebraizing quantification (i.e. for going beyond propositional logics), cf. [RS63, pp. 219–221, 251 (§VI.11)].

75As outlined in §2, most of the approaches in this paper can be viewed as studying algebras whose elements are relations of various ranks, i.e. algebras of relations. E.g. RCA_{n}’s (representable cylindric algebras) are algebras of n–ary relations. From this relational point of view we do not know what the connection is with the RS–approach.

The connection we will indicate goes via algebras of formulas coming from logic.

76More carefully, we would have written Dom(⋂) ⊆ ^{α} (P(U)) where α ≥ ω is basically the same as α was in RCA_{α} at the end of §4. Then ⋂ is an α–ary partial operation on P(U). Then we would write ⋂⟨x_{i} : i < α⟩ or more conventionally ⋂_{i<α}x_{i} (with x_{i} ∈ P(U)) instead of ⋂(X), cf. [RS63, p. 221].

When the approach is applied to nonclassical quantifier logics then BA’s in “BA+⋂” are replaced with the algebraic counterpart of the propositional version of the logic under investigation, i.e. instead of Q–BA’s we investigate Q–Heyting algebras, Q–Post algebras etc., cf. e.g. [RS63, pp. 413–414, 463, 473].

So, what is the connection between this RS–approach and the (more–or–less Tarskian) approaches discussed in this paper and in [HMTII]? Using the terminology of [HMT, §2.7], [JT51], and item (11) below, Q–Boolean algebras are a variant of BAO’s (BA’s with operators) namely they are BA’s with an infinitary partial operator ⋃. Note that ⋃(X) = − ⋂ {−y : y ∈ X}, hence these two partial operations are term–definable from one another. So, a partial algebraic extension of the theory of BAO’s could be applicable here, cf. item (11) below. Pursuing further the connection between Q–BA’s and the Tarskian approach, let us recall the class L ⊆ RCA_{ω} of algebras of finitary relations discussed below Theorem 5 in §4. Let us see if RCA–operations are expressible in Q–BA’s or if the Q–BA operation ⋂ is expressible in RCA’s. In L we have c_{i}x = ⋃ {s_{j}^{i}x : j ∈ ω} (where ⋃ denotes supremum), moreover, in those members of RCA_{α} (α ≥ ω) which are called weak cylindric set algebras, under rather mild conditions c_{i}x = ⋃ {s_{j}^{i}x : j ∈ α} where ⋃ is set theoretic union. (These mild conditions amount to assuming that every element of the algebra is a fixed point of one of the c_{j}’s and that there is a sequence q in the greatest element 1 of the algebra such that for any other sequence f in 1 we have Rng(f) ⊆ Rng(q).) More generally applicable ways of expressing cylindrifications (the c_{i}’s) in terms of ⋂ and s_{j}^{i} are in [HMTII, p.268 item (K_{1})], in [Pr70], [P73a]. We do not know how to express the s_{j}^{i}’s in terms of Q–BA’s. In the other direction, let 𝔄 ∈ L. Let ⋃ X be defined iff X = {s_{j}^{i}y : j ∈ ω} for some y ∈ A and i ∈ ω. Further let ⋃ {s_{j}^{i}y : j ∈ ω} = c_{i}y. Letting 𝔄^{+} be the partial algebra ⟨A, ∨, ∧, −, s_{j}^{i}, ⋃⟩_{i,j∈ω}, the operations of the cylindric algebra 𝔄 are term definable in 𝔄^{+} (as indicated above). If we omit the s_{j}^{i}’s from 𝔄^{+} then we obtain a Q–BA (but then we do not know how to recover 𝔄^{+}).

This kind of algebraic logic has profound applications in various areas, e.g. in pure logic, in set theory, and in computer science, cf. Scott [Sco66,Sco68, Sco70], Moschovakis [Mo73], Mirkowska [Mi77], [Ba77], Karp [Ka64]. Handbook of Math. Logic p.1024 extensively discusses such applications. We note that Scott [Sco66] gives a relatively simple direct proof of the independence (from ZFC) of the continuum hypothesis without using special set theoretical methods like forcing. Instead of forcing he uses this part of algebraic logic.

In the present survey we cannot attempt giving an adequate review of this branch of Algebraic Logic, but for an incomplete list of further references see the works of Rasiowa (and her co–authors) in the References, and also [Ms77, Ms79, Ms88], [MsR74], [BB73], [BrS73], [G85], [GP89], [GgP88], [FV89–91], [Te86].

(8) Connections with, and approaches originating from, computer science

This area was sporadically mentioned throughout this paper, we do not repeat those references. For lack of space, here we restrict ourselves to a list of references the items of which are not listed in [HMTII]. Even if we restrict our attention to papers that appeared after [HMTII], our list is severely incomplete. Some of the recent works in this area are Cosmadakis [C87], Trnková– Reiterman [TR87], [MP88], [C88a, C86], Imielinski–Lipski [IL84], Venema [V89], [W88], [L87], [LM89], [GL90], [Ca88], Goldblatt [G87], Jónsson [J89, J90], Salibra [Sa89], Ng [N84], Volkov [Vo86– 88a], [N80, N81], Sanders [Sa80], Knuth–Rónyai [KR83], [BZ86], [BKSS], van Benthem [vB88–91], [AGN87], [C87], [O89, O89a], Schönfeld [Sö77], Düntsch [Dü89], the textbook [SS89], [BMP], Pratt [Pr90], [K90], [N90], [H80], [HJ86], [BU77], [AN86], [Pl88, Pl89], [Ro72], [Zi83], [Bn89], [Co91a], [V91], Ibrahim–van Emden [IE91], papers in the Handbook of Theoretical Computer Science vol. B, e.g. [Ka90] (e.g. p.1085, 1083).

(9) Connections with category theoretical logic

First we briefly return to the finitization problem using the language of category theory. Let α be an ordinal (as before). Let Full Cs_{α} be the usual category of full Cs_{α}’s (see §4) as objects and homomorphisms as morphisms between them. Let SetIso be the category of sets as objects and isomorphisms (bijections) as morphisms between them. Then there is a functor F_{α} : SetIso ↣ Full Cs_{α} such that F_{α}(U) = ℜ𝔢𝔩_{α}(U) = ⟨𝔓(^{α}U), …⟩, and for any morphism p : U ↣ → W, F_{α}(p) : P(^{α}U) ↣ → P(^{α}W) is the natural one, i.e. F_{α}(p)(R) = {q ∘ p : q ∈ R} for R ⊆ ^{α}U. It is not hard to check that F_{α} is really a functor (cf. Exercise 3P (Algebraic Logic) (c) in [AHS]). (In [HMTAN], [HMTII, p. 15], F_{α}(p) was called the base isomorphism induced by p, and F_{α}(p) was denoted by p̃^{α}.)

Let K^{+} be an expansion of the class full Cs_{α}, i.e. full Cs_{α} = Rd(K^{+}), cf. Remark 2. We say that K^{+} is functorially invariant if we can “expand” F_{α} defined above to a functor F_{α}^{+} : SetIso ↣ K^{+} such that F_{α}(U) = Rd(F_{α}^{+}(U)), and F_{α}(p) “as a function” coincides with F_{α}^{+}(p). Here K^{+} is the category made from K^{+} the usual way (adding the usual homomorphisms as morphisms).^{77}

As noted in a footnote in Remark 2, functorial invariance is stronger than invariance (the latter quoted in Remark 2 from the literature), and conforms more to algebraizing abstract model theory. (The finitely axiomatizable expansion of RQA_{ω} in Sain [S87a] and in Theorem 16 in §6 is functorially invariant, too.)

PROBLEM 3. Find functorially invariant K^{+} such that SPK^{+} is a finitely axiomatizable variety (or at least a finitely axiomatizable quasivariety) satisfying one of (i)—(iii) below.

(i) full Cs_{α} = Rd(K^{+}) for α = ω or for 2 < α < ω.

(ii) The same as (i) but with full set RQA_{α} in place of full Cs_{α}.

(iii) The same as (i) but for Pinter’s substitution–cylindric algebras (see §5).

The above problem comprises 6 different problems, each one interesting on its own right, e.g. the α < ω and α = ω cases are two separate problems.

For α = ω, (ii), (iii) above were provided with one solution in Sain [S87a], but it would be still immensely interesting to see simplifications (both of K^{+} and of the axiomatization of the variety generated by K^{+}) of that solution, and to see different solutions. It would be also very interesting to see a (preferably streamlined) finite axiomatization of the quasivariety generated by {F_{α}^{+}(U) : U is a set} for the choice of K^{+} given in [S87a] (as a solution of (ii), (iii) above), where the functor F_{α}^{+} : SetIso ↣ K^{+} was introduced preceding Problem 4.1.

Connections with category theoretical logic were sporadically mentioned throughout this paper. E.g. it was pointed out that RRA’s are actually categories. We should mention that so are the various kinds of algebras unifying all finitary relations (RCA_{ω}’s, RQA_{ω}’s etc.) discussed in this paper. The objects of these categories are the elements representing ^{n}U for n < ω. This is the easiest to see in the case of many–sorted cylindric algebras in subsection (1) above. There the objects of the category are the sorts of the algebra. Then elements of the algebra can be considered as relations acting between sorts, and in this quality they are morphisms acting between the sorts as objects.

Another, more generally used approach (cf. [Z83], [Z84], [D69], [Mk87], [MkR77]) considers all elements of a many–sorted cylindric algebra 𝔄 (or a partial cylindric algebra, cf. items (1,2) above) as objects of the corresponding category C. A pair ⟨R, S⟩ of elements R, S of 𝔄 is a morphism of C iff R can be considered as a function with range contained in S (i.e. whenever S is an n–ary relation then R is an n + k–ary one for some k, and can be considered as associating n–tuples to certain k–tuples; the codomain of this morphism is S, while its domain is c_{n}^{−}c_{n+1}^{−} ⋯ c_{n+k}^{−}(R)). (This connection extends to ordinary cylindric algebras the obvious way, i.e. by transforming 𝔄 into such an algebra.) If we turn a cylindric algebra 𝔄 into a category C along these lines then any functor F : C ⟶ Set preserving^{78} finite limits and finite colimits will correspond to a model of the usual first–order theory represented by 𝔄 or, equivalently, F will correspond to a representation of (a subdirectly indecomposable factor of) 𝔄 as a (cylindric) set algebra or, still equivalently, F will correspond to a homomorphism from 𝔄 into a (cylindric) set algebra. Roughly speaking, the special categories called small pretoposes (see e.g. [Mk87], [MkR77]) in categorical logic correspond to (“abstract”) cylindric algebras (or RQA_{ω}’s), and pretopos functors from small pretoposes into Set correspond to homomorphisms from arbitrary cylindric algebras into cylindric set algebras (Cs_{ω}’s or Qps_{ω}’s). The large pretopos Set corresponds to the class of full Cs_{ω}’s. The reason why there is no single Cs_{ω} (but instead a class of them) corresponding to Set is that Set is a proper class, and for “purely administrative” reasons, algebras are required to be sets. It is not hard to construct e.g. a single class size Qps_{ω} 𝔅 such that 𝔅 would play the rôle of Set in quasi–polyadic algebraic logic. In particular, the class of all models of a theory T would correspond to the class Hom(𝔄_{T} , 𝔅) of homomorphisms from the RQA_{ω} 𝔄_{T} corresponding to T into our single “class–algebra” 𝔅.

77Using the terminology of [HMTAN], [HMTII], K^{+} is functorially invariant if its full members “admit” base– isomorphisms the same way as those of Cs_{α} do. In passing we note the following. If we want F_{α} to have an inverse, say G_{α} : Full Cs_{α} ↣ SetIso, then restricting our attention to full algebras is important here. Namely, Andréka constructed a simple RRA 𝔄 and an automorphism h of 𝔄 such that in no representation of 𝔄 is h induced by a permutation of the base set U (of the algebra 𝔅 ⊆ ⟨𝔓(U × U) …⟩ representing 𝔄). I.e. for any embedding of 𝔄 into a full set RA ⟨𝔓(U × U) …⟩, h does not extend to an automorphism of the full algebra. Her construction carries over to cylindric and polyadic algebras from RA’s. This solves Problem 19a of Maddux [Ma89c]. Shelah proved that similar kinds of results can apply even for very innocent looking cylindric algebras [Sh89], [BiS88] (solving a problem from [HMTAN]).

We are afraid that the remarks made so far do not convey enough of the subject of this item. For lack of space we restrict ourselves to giving an incomplete list of references.: Daigneault [D69], Monk [M78], Zlatos [Z83, Z84], Comer [Co72], Barthelemy [B74], Georgescu [G73], Gergely [Ge80], Jónsson [J88], Olivier–Serrato [OS80], Laita [L76], Ouellet [O82], Volger [V75], Topencarov [T74], Freyd–Scedrov [FS90], Sain [S89].

The natural category theoretic investigations (which have already been carried through in great detail for groups, semigroups etc.) for cylindric and related algebras constitute an area where more work would be welcome. (E.g. the connections between RRA’s and RCA_{n}’s in [HMTII, §5.3] are not functorial in their present form, and it would be very nice to have such a version.) For fragmentary results and open problems see e.g. Preller [Pr68], Adámek–Herrlich–Strecker [AHS], and Sain [S89].

(10) Peano Arithmetic, Gödel’s incompleteness etc.

The algebraic approaches (sometimes called multi–modal logic approaches) to this field use BAO’s (Boolean algebras with operators) just as the rest of the directions reported herein. Usually at least one of the extra–Boolean operators is based on self–reference as the latter was used in Gödel’s proof (definability of syntactic concepts in strong enough theories). This field became recently very active. See Solovay [So76], Smorynski [Sm85, Sm84], and the references in Chapter 2 of Bernarducci [B89]. See also Halmos [Ha60], and [Ha85, pp. 206–215], [M77, Problems 1, 4, 5]. (Works of Magari, Mangione, Hansoul, and Jeroslow quoted in [HMTII] are relevant to the present subject.) For a sample of works too recent for being quoted in [HMTII] on BAO’s, and their strong connections with multi–modal logics (temporal logics) see e.g. Goldblatt [G85, G88, G87], Naturman–Rose [NR89], Bernardi–D’Aquino [BD88], Brink [Bn88], [Sha80], Blok [Bl80], [Bl80a], [Ra79], [Te86], and the references in Remark 1 in §3 herein.

78Set is the usual category of sets (as objects) and functions between them (as morphisms).

(11) Boolean algebras with operators (BAO’s)

As already mentioned e.g. in Remark 1 (§3), the theory of BAO’s is an important one unifying not only practically all the approaches discussed in this paper, but a large portion of propositional algebraic logic, too. Besides in algebraizations of quantifier logics, BAO’s play a fundamental rôle in the theories of modal, multimodal, and temporal logics, cf. e.g. Goldblatt [G76–G89]. To mention only one example, what are known today as Kripke–models, were first discovered in BAO–theory, cf. Jónsson–Tarski [JT48], [JT51].^{79}

Some of the references are [HMT, §2.7], [JT51], Jónsson [J84, J90, J91], Henkin [H70], Goldblatt [G85, G87, G88, G89], Andréka–Givant–Németi [AGiN93], Naturman–Rose [NR89], Bernardi– D’Aquino [BD88], Brink [Bn86–91], [Sha80], Blok [Bl76, Bl80, Bl80a], Blok–Pigozzi [BP89a, BP89b], [Ra79], [Te86], Sain [S84], [N80], works of Hansoul and Wu in the bibliography of [HMTII].

Non–classical (i.e. non–Boolean) generalizations of BAO’s: If we want to study e.g. intuitionistic multi–modal logic then we have to generalize Boolean Algebras with Operators accordingly. Indeed, if we replace the Boolean part of BAO’s with the algebras of some nonclassical propositional logic we will obtain standard generalizations of BAO’s like Heyting algebras with operators ([M60], reference Fischer–Servi [81’] of [HMTII], [FV89], [F87], [Ve86]), De Morgan algebras with operators, Hilbert algebras with operators, distributive lattices with operators ([Co89a], [IL84], [A89a], [A89],[G89], R.J. Thompson presented new results on this in a seminar series in Berkeley 1991). Weak Brouwerian semilattices with operators and pseudo– interior algebras with operators were proved in [BP89b] to be of central importance for the theory of discriminator varieties. See also [Sw80] and references Georgescu [72’], Servi [79’] of [HMTII]. Some classes of Heyting algebras with operators suitable for studying certain intuitionistic modal logics were investigated in [FV89].

\underline{Remark.} If Boolean (unary) complementation is not available in our generalized BAO’s (as is the case with e.g. distributive lattices with operators), then it is important to add the duals c_{i}^{∂}(x) = −c_{i} − (x) of our operators c_{i} (or explain why they are not needed). Carrying this principle through e.g. on the algebras in the above quoted works is a research program which remains to be carried through in the future.

(12) Logics without variables (combinatory logic)

This fairly active branch of logic is strongly related to algebraic logic. Both fields (logic without variables, LWV for short, and algebraic logic, AL) aim for a more refined analysis of the structure of formulas (together with their meanings, proofs etc.) than conventional logic does. Namely, conventional logic takes atomic formulas with all their substituted versions like R(v_{0}v_{1}), R(v_{1}v_{0}), R(v_{2}v_{2}) etc. for granted, i.e. they are treated as “primitives”. A disadvantage of the conventional view is that these primitives are far from being independent of each other, and in particular they cannot be replaced by each other in formulas without affecting validity. A consequence of this conventional treatment of atomic formulas, often mentioned both in LWV and AL, is a certain complication in the axioms or proof rules when one has to talk about free and bound occurrences of individual variables when specifying an axiom or rule. A typical example is the axiom schema saying “∃v_{i}φ → φ if v_{i} does not occur freely in φ”. This kind of complication (reference to free variables) is not acceptable in propositional logics. Both AL and LWV eliminate this complication from quantifier logics. This is obviously true for LWV, concerning AL we mention that the inference system for predicate logic given in [BP89, App C, p. 69], no axiom or rule schema refers to individual variables. Actually, in conventional logic, talking about free occurrences of variables is needed to express the connection between the atomic formulas R(v_{0}v_{1}), R(v_{2}v_{2}) etc. mentioned above. Therefore, if we want to get rid of variables, we will have to express these connections by some other means, e.g. by new logical connectives. Indeed, both AL and LWV begin eliminating (explicit mention of) free individual variables by introducing new logical connectives binding the substitution instances of R(v_{0}v_{1}) to each other. Such a connective is p_{01} = S_{[0,1]} recalled in §5 herein, and discussed throughout this paper. Since p_{01}R(v_{0}v_{1}) is logically equivalent with R(v_{1}v_{0}), we may restrict our supply of atomic formulas to the ones of the form R(v_{0}v_{1}) i.e. to the ones in which the individual variables occur in a once for all fixed order v_{0}, v_{1}, …, v_{n}, v_{n+1} … if we have all the p_{ij}’s and s_{j}^{i}’s as logical connectives. But since the individual variables following predicate letters are uniquely determined, one can just as well dispense with variables altogether. Indeed, this is done in many of the LWV papers as reviewed at the end of §I in Kuhn [Ku80]. In many of the LWV papers, the logical connectives p_{ij}, s_{j}^{i} are called combinators or combinatory predicate functors, see Došen [Do88, item (2), p. 249]. In the AL literature, the above indicated way of eliminating variables is described in detail e.g. in [AGN77], [HMTII] (restricted formulas), [BP89, App C], Sain [S87a, §4], to mention a few recent sources; but this is a classical tool of AL.

79In passing we note that Magari’s school does not seem to be aware of the presently reported research direction (i.e. BAO’s) as something that was started by Jónsson and Tarski in 1948.

To indicate why AL would want to get rid of the conventional machinery of free individual variables, observe that when translating an algebraic equation like x ≤ c_{0}x to logic, we obtain the logical schema φ → ∃v_{i}φ in which the free variables of φ are not mentioned. The algebraic variable x translates to a formula φ or to a metavariable φ running over formulas. What it does not translate to is an individual variable. Somehow the individual variables of logic (sometimes indicated in logic as φ(v_{1}v_{2}) meaning that v_{1} and v_{2} are free variables in φ) do not show up on the algebraic side. Very roughly speaking, this is one of the reasons why AL and LWV are so strongly connected. (Another reason is that they both are ambitious analyses of logic, and they both are deeply concerned with finding the right concepts.)

A very incomplete list of references is: Došen [Do88], Quine [Q36, Q71, Q81], Kuhn [Ku80, Ku83], [Bc85], Venema [V89, V90, V91], Craig [Cr74], Tarski–Givant [TG87], [Gi89], [Si90, Si91].

Further discussion of the connection between combinatory logic and AL is found in [Q71], and in the excellent recent book [TG87]. [TG87] is of central importance in understanding the deep connections between Tarskian AL and Tarski’s approaches to other foundational fields. A very long time was devoted to the careful and ambitious preparation of that book. Indeed, it provides an illuminating synthesis of the various schools of thought (think of “Tarskian semantics” as an example) Tarski created or pioneered.

(13) Connections between algebras of logic and “pure logic”; universal algebraic logic

In a sense, these investigations concern the connections between algebraic logic (AL) in the narrower sense on the one hand, and logic (mathematical logic, philosophical logic, symbolic logic etc.) in itself on the other hand. (This sentence is somewhat paradoxical because it might seem to imply that AL does not contain these connections, and that logic in itself does not contain AL, but no such implication was intended here.) One of the main concerns is explicit applications and interpretations of results of AL, in “pure logic”. (The other direction is not ignored either.)

Universal algebraic logic starts out from a very general definition of a logic L (of which the well known logics like classical logic, modal logics, temporal logics, intuitionistic logic, computer science logics etc. are special cases), and investigates the kinds of mathematical objects (e.g. classes of algebras like that of cylindric algebras, multi–modal algebras, Heyting algebras etc.) associated to L by AL. Associating such objects to L is called algebraization of L. One of the subjects investigated here is elaborating a general method for algebraizing L ([HMTII, pp. 255–260]), another is finding an adequate criterion for algebraizability of L (Blok–Pigozzi [BP89]) which was called “adequateness criterion” in [AS78] and “semantical well presentability” in [HMTII, p. 257], [ANS84]. The recent [FV91], [FV90] point in the direction of possible unifications of algebraizability in the sense of [BP89, BP89c, BP89d] and that in the sense of [HMTII, §5.6], [AN78], [ANS84]. A typical problem area of universal AL is proving that (independently of the choice of L) a certain logical property of L is equivalent with an algebraic property of the corresponding class of algebras (cf. the works of Blok and Pigozzi, [HMTII, §5.6], [ANS84], [AGN73, AGN77], [AN75, §1], [O86], [Ma77], [W84], [S89]).

Algebraic model theory concentrates on the algebraization of L from the point of view of the model theory of the logic L. (This again might sound somewhat paradoxical since the model theory of L might already have some algebraic flavor. In this area, however, algebraization is independent of that algebraic flavor.) Cf. [HMT, §4.3], [N89, 78], [S89], [Se85, 86], [Sh89], [J73]. For lack of space we cannot discuss the present subject (connections between AL and logic) adequately. Further references are e.g. [S87, §4], [Bl76–BP89], [Co72], [Cz79], [G76], [M71, 77, 78], [N85a, 87], [O89], [P89].

(14)

For algebraizations of quantifier logics not mentioned in this paper the reader is referred to Chapters 5, 6 of [HMTII] (cf. also §7 of [M77]).

§8. APPENDIX 1: The traditional finitely axiomatizable approximations CA, RA etc. of our classes RCA, RRA etc. of algebras of relations

Definition 8.1. By an α–ary cylindric Kripke frame we understand a relational structure 𝔚 = ⟨W, ≡_{i}, E_{ij}⟩_{i,j<α} where E_{ij} ⊆ W and ≡_{i} is an equivalence relation on W satisfying (1–4) below, for all i, j < α.

(1) (≡_{i} ∘ ≡_{j}) = (≡_{j} ∘ ≡_{i}), i.e. the equivalence relations commute;

(2) E_{ij} = E_{ji} is a set of representatives for ≡_{i}, i.e.

(∀u ∈ W)|(u/ ≡_{i}) ∩ E_{ij}| = 1 if i ≠ j, else E_{ii} = W;

(3) (∀u ∈ E_{ij})(u/ ≡_{k}) ⊆ E_{ij} for k ∉ {i, j}, i.e. E_{ij} is a union of ≡_{k}–equivalence classes;

(4) E_{ij} ∩ E_{jk} ⊆ E_{ik}.

Let 𝔚 = ⟨W, ≡_{i}, E_{ij}⟩_{i,j<α} be as in Definition 8.1. The powerset algebra or complex algebra 𝔓(𝔚) of 𝔚 is defined as follows:

𝔓(𝔚) = ⟨𝔓(W), C_{i}, E_{ij}⟩_{i,j<α}

where C_{i}(X) = ⋃ {(u/ ≡_{i}) : u ∈ X}, i.e. C_{i}(X) consists of the ≡_{i}–neighbors of elements of X.

Definition 8.2. The class CA_{α} of α–ary cylindric algebras is defined as

CA_{α} = S {𝔓(𝔚) : 𝔚 is an α–ary cylindric Kripke frame} .

Recall that S is understood up to isomorphism.

Proposition 8.3. (i) CA_{α} is a variety axiomatizable by a finite schema of equations. (ii) CA_{n} is axiomatizable by a finite set of equations, for n < ω.

Outline of proof: Translating the conditions in Definition 8.1 (making 𝔚 a cylindric Kripke frame) into equations valid in 𝔓(𝔚) is an easy exercise. E.g. transitivity of ≡_{i} translates to the equation c_{i}c_{i}x ≤ c_{i}x; or reflexivity of ≡_{i} translates to x ≤ c_{i}x, while symmetry translates to c_{i} − c_{i}x ≤ −x. A system of equations obtained this way is the following, postulated for all i, j, k < α.

(E0 ) the Boolean equations  (E4 ) c_{i}c_{j}x = c_{j}c_{i}x
(E1 ) c_{i}(x ∨ y) = c_{i}x ∨ c_{i}y  (E5 ) Id_{ii} = 1 and Id_{ij} = Id_{ji}
(E2 ) x ≤ c_{i}x  (E6 ) Id_{ik} = c_{j} (Id_{ij} ∧ Id_{jk}) if j ∉ {i, k}
(E3 ) c_{i} − c_{i}x = −c_{i}x  (E7 ) Id_{ij} ∧ c_{i} (Id_{ij} ∧ x) ≤ x if i ≠ j.

Claim 8.3.1 CA_{α} coincides with the variety defined by (E0–7).

To prove this: First observe that (E3) ⊧ c_{i}c_{i}x = c_{i}x as follows, writing c instead of c_{i}. ccx = c − −cx = c − c − cx = −c − cx = − − cx = cx. It is easy to check that CA_{α} ⊧ (E0–7); (E0–4) correspond to the first part of Definition 8.1 including (1), while (E5–7) correspond to (2–4). In particular, (E7) together with c_{i}Id_{ij} = 1 correspond to the main part of (2), where c_{i}Id_{ij} = 1 follows from (E5–6). Therefore, what remains to prove is that every algebra in which (E0–7) is valid is in CA_{α}. This goes by repeating the proof of the Stone representation theorem for Boolean algebras, but now paying attention to the non–Boolean operations (c_{i}, Id_{ij}) too. This is not very hard, and is described in detail e.g. in [HMT, 2.7.4–5, 2.7.40]. Terminology: We say that CA_{α} is finitizable exactly because Proposition 8.3 holds; namely because both CA_{n} is finitely axiomatizable and CA_{α} is axiomatizable by a finite schema, for α ≥ ω.

Proposition 8.4. RCA_{α} ⊆ CA_{α}.

Proof: By Proposition 8.3 it its enough to prove that fullCs_{α} ⊆ CA_{α}. Let 𝔄 = ⟨𝔓 (^{α}U) , c_{i}, Id_{ij}⟩_{i,j<α}. For any p, q ∈ ^{α}U let p ≡_{i} q iff (∀j ∈ α ∖ {i})p_{j} = q_{j}. It is easy to check that 𝔚 = ⟨^{α}U, ≡_{i}, Id_{ij}⟩ is a cylindric Kripke frame. Since 𝔄 = 𝔓(𝔚), we are done.

Summing up, we have a finite schema axiomatizable variety CA_{α} containing RCA_{α} as a finitary approximation of RCA_{α}. The next natural question asks how good this approximation is, how far CA_{α} is from RCA_{α}. The representation results in [HMTII, §3.2] seem to say that they are pretty close. For example a result says that if a CA_{α} is atomic with all atoms “rectangular” then it is already in RCA_{α}, where an atom x of a CA_{2} is rectangular iff x = c_{0}(x) ∧ c_{1}(x). Rectangularity for arbitrary α is the natural generalization of this (the intuitive idea being that the rectangular “relation” x is a Cartesian product), cf. [HMT, §1.10, p. 227]. Another result says that if 𝔄 ∈ CA_{α} and (∀x ∈ A)(∃ infinite H ⊆ α)(∀i ∈ H)c_{i}x = x then 𝔄 ∈ RCA_{α}.

By Monk’s non–finitizability result (Theorem 4 here) the gap will always remain infinite between RCA_{α} and its finitary approximation CA_{α}. The same applies to BRA’s, RRA’s etc.

To illustrate the difference between RCA_{α} and CA_{α}, the figure below represents a cylindric Kripke frame 𝔚 for which 𝔓(𝔚) ∈ CA_{2} ∖ RCA_{2}. W has 5 elements, the horizontal, solid line represents ≡_{1}, while the vertical broken one represents ≡_{0}. E.g. d ≡_{0} e and d ≡_{1} e, further d ≡_{0} a but d ≢_{1} a.

E_{01}
b  c

a  d
e

𝔚

FIGURE 5

Another example is the following^{80}:

80This example was found by J. Madarász.

E_{01}
b  c

a  d
e

𝔙

It is easy to check that 𝔚 satisfies the conditions in Definition 8.1, hence 𝔓(𝔚) ∈ CA_{2}. It is not hard to see that 𝔓(𝔚) ∉ RCA_{2} since 𝔓(𝔚) is simple, hence if it were in RCA_{2} then it were isomorphic to a Cs_{2} etc. We omit the rest but the reader is invited to try to represent 𝔓(𝔚) as a Cs_{2}.

∗ ∗ ∗ ∗ ∗

The finitely axiomatizable variety RA of relation algebras approximating RRA is obtained the following way. Let (3.1)^{+} be obtained from (3.1) above Theorem 1 in §3 by replacing “involuted semigroup ⟨A, ∘,^{−1}⟩” in (3.1) with “involuted monoid ⟨A, ∘,^{−1}, Id⟩”. Consider the equation

(8.1)  x^{−1} ∘ −(x ∘ y) ≤ −y.

An algebra is an RA iff it satisfies both (3.1)^{+} and (8.1). It is easy to see that {(3.1^{+}), (8.1)} is a finite set of equations, hence RA is a finitely based variety. Clearly, RA ⊇ RRA.

Completely analogous observations apply to the RA–RRA relationship as were outlined for the CA–RCA relationship above.^{81}

Remark (in connection with defining RA). Let RA^{−} be the variety defined by (3.1)^{+}. Then RA^{−} ⫌ RA, moreover RA^{−} is not a discriminator variety (while RA, CA_{n} etc. are such). What is lost in RA^{−} is that if we define c_{0}x as 1 ∘ x then though c_{0} remains a closure operator in RA^{−}, it is no more a complemented one, i.e. the complement of a c_{0}–closed element need not be c_{0}–closed.

Therefore, if we take [HMTII, Def.5.3.1, p. 211] defining RA, and remove the last equation (R_{7}) then the variety defined by the remaining postulates (R_{1} − R_{6}) is not discriminator. (We note that (R_{7}) is our (8.1), and (R_{1} − R_{6}) is equivalent with our (3.1)^{+}.) A similar remark applies to other standard definitions of RA, e.g. to [JT52, Def.4.1].

If we add the following simple equation

(8.2)  1 ∘ −(1 ∘ x) ≤ −x

to (3.1)^{+} then we obtain a discriminator variety containing RA. (Actually the weaker equation 1 ∘ − (1 ∘ [x ∨ x^{−1}]) ≤ −x is sufficient in place of (8.2).)

It is useful to know whether we are in a discriminator variety e.g. because in any discriminator variety V, quasi–equations and equations are equivalent in the sense that any quasi–equation q can be translated to an equation e such that for every subdirectly irreducible 𝔄 ∈ V, (𝔄 ⊧ q iff 𝔄 ⊧ e). So as soon as we know that we are in a discriminator variety, we may use quasi–equations in defining subvarieties.

81See e.g. the results quoted from Maddux [Ma87a, 87] etc. close to the end of §3. Further, Givant proved that if an RA 𝔄 is generated by a chain H ⊆ A of equivalence elements then 𝔄 is representable. This generalizes a result of Jónsson. Givant generalized parts of this to cylindric algebras.

Maddux [Ma78, Ma82, Ma87] introduced the important varieties NA ⊃ WA ⊃ SA ⊃ RA containing RA. These are obtained by weakening associativity of “∘” in (3.1)^{+}. These are important from the point of view of algebraizing syntactic aspects of logic (e.g. proof theory) as well as from the point of view of the so called relativized algebras of relations, cf. the discussions around Theorems 6, 7. Sain proved that in WA and NA, for any class K with

“simple algebras” ⊆ K ⊆ “subdirectly irreducible algebras”,

K is not first–order axiomatizable (actually, K ≠ UpK), solving a problem of Maddux [Ma78, §8, p. 112], or [Ma89c, Problem 5]. (Sain proved the same for the cylindric algebraic counterpart of these classes [cf. e.g. the discussions around Theorems 6, 7] e.g. for ICrs_{α}, too.) The proof uses methods in [S84, S89]. These results show that WA and NA are far from being discriminator, since in discriminator varieties every such K is axiomatizable by a single universal formula. (SA is a discriminator variety.)

∗ ∗ ∗ ∗ ∗

The class RQPA_{α}, abbreviated as RQA_{α} in the present paper, of representable quasi–polyadic algebras of α–ary relations was discussed in §5. The finitizable variety QPA_{α} (of quasi–polyadic algebras) approximating RQPA_{α} is defined analogously to CA_{α} or RA by a finite schema Σ_{0} of equational axioms. This approximation Σ_{0} of QPA_{α} was discussed and outlined above and below Theorem 12 in §5.

So far we discussed the finitizable approximations CA_{α}, RA, and QPA_{α} of our classes RCA_{α}, RRA, and RQA_{α} of algebras of relations. The finitizable approximations, PA_{n}, PEA_{n} etc. of our other kinds RPA_{n}, RPEA_{n} of algebras can be defined analogously to our above definitions (of CA_{α} and RA), and more or less analogous observations can be made. There is a general notational convention: the finitary approximating class has the shorter name (like CA_{α} or RA), and the name of the approximated class is obtained by writing an “R” in front of the short name (like RCA_{α} or RRA). Here this extra R stands for representable.

For completeness, we recall^{82} below the axiomatic approximations QPA_{α} and SC_{α} of RQPA_{α} and RSC_{α}. (As we said before, these correspond to first order logic without equality.)

Consider the following equations (E8-E16).

(E8) s_{i}^{i}(x) = p_{ii}(x) = x. (E9) s_{j}^{i} and p_{ij} are Boolean endomorphisms. (E10) p_{ij}x = p_{ji}x. (E11) s_{j}^{i}c_{i}(x) = c_{i}(x). (E12) c_{i}s_{j}^{i}(x) = s_{j}^{i}(x) whenever i ≠ j. (E13) s_{j}^{i}c_{k}(x) = c_{k}s_{j}^{i}(x) whenever k ≠ i, j. (E14) p_{ij}p_{ij}(x) = x. (E15) p_{ij}p_{ik}(x) = p_{jk}p_{ij}(x) whenever i, j, k are distinct. (E16) p_{ij}s_{i}^{j}(x) = s_{j}^{i}(x).

Now, SC_{α} is defined by (E0–E4, E8–E9) omitting all references to p_{ij}’s, of course.

SC_{α} = Mod(E0 − E4, E8 − E9), as we just said,
QPA_{α} = Mod(E0 − E4, E8 − E16), and
QPEA_{α} = Mod(E0 − E16, s_{j}^{i}(x) = c_{i}(Id_{ij} ∧ x)).

The finitizable approximating varieties CA_{α}, RA, QPA_{α}, PA_{n}, QPA_{n} etc. are discussed in the two volume monograph [HMT], [HMTII], and in the references therein. CA_{α} is studied in especially great detail: almost the whole of the first volume [HMT] is devoted to CA_{α}.

82From Sain–Thompson [ST91], cf. also [AGMNS]Def.4.1.

Remark (discriminator varieties). CA_{n}, RA, PA_{α}, PEA_{α}, QPA_{n}, QPEA_{n} are all discriminator varieties (n ∈ ω and α an arbitrary ordinal). Actually, CA_{n} is contained in a discriminator variety V_{n} much larger than CA_{n}. Namely, let c_{(n)}x =^{def} c_{0} … c_{n−1}(x). Then V_{n} is defined by the Boolean axioms taken together with the following 4 equations:

x ≤ c_{(n)}x, c_{(n)}0 = 0,

c_{i}(y ∧ c_{(n)}x) ∧ c_{(n)}x = c_{i}(y) ∧ c_{(n)}x,
c_{i}(y − c_{(n)}x) − c_{(n)}x = c_{i}(y) − c_{(n)}x, for all i < n; where (x − y) abbreviates (x ∧ −y).

Clearly CA_{n} ⫋ V_{n}. The discriminator term for V_{n} is the one given in the proof of Theorem 3. The idea of the proof is that the above equations ensure that “relativizing” both with c_{(n)}x and −c_{(n)}x are homomorphisms, hence in any subdirectly irreducible member of V_{n} we have c_{(n)}x ∈ {0, 1}. Note that the above equations do not tell us anything about the behaviors of the Id_{ij}’s. An immediate corollary of this is that the (fairly elaborate) axioms governing the Id_{ij}’s (recall Definition 8.1 (2–4)) do not contribute to CA_{n}’s being a discriminator variety. (We note that we could achieve even bigger discriminator varieties by picking some cylindric algebraic term σ(x) and writing σ(x) in place of c_{(n)}x everywhere in the above equations. Of course, one has to choose σ(x) in such a way that these equations remain valid in CA_{n}.)

§9. APPENDIX 2: Universal Algebraic Background

Usually, it is a difficult task to prove that a certain class of algebras is closed under homomorphisms. Theorem 9.2 below gives us considerable help by proving that certain quasivarieties are already varieties. Definition 9.1 and a part of Exercises 9.1 below have already been given in §3, in the “universal algebraic background” material preceding the proof of Thm.1. For smoother readability of the present appendix, we include them here, too. (We note that §3 contains more exercises than the present section.)

Notation: For any class K of similar algebras Qeq(K) denotes the set of all quasiequations in the language of K which are valid in K. Thus ModQeq(K) is the class of all models of the theory Qeq(K). Hence ModQeq(K) is the quasivariety generated by K.

The following is one of the basic theorems of universal algebra. It is called the characterization of quasivariety generation. (Its analogue for varieties is the well–known Birkhoff’s variety theorem, cf. e.g. Buris–Sankappanavar [BS81] or Monk [M76].)

THEOREM 9.1. For any class K of similar algebras

ModQeq(K) = SPUp K.

To prove Theorem 9.1 we will use the claims below. Recall that for an arbitrary class K of algebras,

P^{r}K =^{def} I {Π_{i∈I}𝔄_{i}/F : F is a filter over the set I, and (∀i ∈ I)𝔄_{i} ∈ K} .

The following is an easy fact of elementary universal algebra (cf. also e.g. Burris–Sankappanavar [BS81] or Németi–Sain [NS81]). Claim 9.1.1: For any class K of similar algebras

P^{r}K ⊆ SPUp K.

Proof of Claim 9.1.1: Let I be a set, {𝔄_{i} : i ∈ I} ⊆ K, F a filter over I, 𝔄 =^{def} Π_{i∈I}𝔄_{i}/F ∈ P^{r}K. Let

U =^{def} {G : G an ultrafilter over I and G ⊇ F}.

Let h : 𝔄 ⟶ Π⟨Π_{i∈I}𝔄_{i}/G : G ∈ U⟩ be defined as h(a/F) =^{def} ⟨a/G : G ∈ U⟩. It is not hard to check that h is an embedding, therefore 𝔄 ∈ SPUp K.

Claim 9.1.2: For any class K of similar algebras (i) SS K = SK (ii) P^{r}S K ⊆ SP^{r} K (iii) P^{r}P^{r} K = P^{r}K.

Proof of Claim 9.1.2: The proofs are easy, since they use only the definitions, cf. e.g. Burris– Sankappanavar [BS81] Lemma 2.22.

Claim 9.1.3: For any class K of similar algebras

K = ModQeq(K) ⟺ K = SK = P^{r}K.

Proof of Claim 9.1.3: It is relatively easy, cf. e.g. Burris–Sankappanavar [BS81] Thm.2.25, or Németi–Sain [NS77], Andréka–Németi [AN76], Sain [S94].

Proof of Theorem 9.1: First we prove that for any class K of similar algebras SP^{r} K = SPUp K holds. Indeed,

(+)  SPUp K ⊆ SP^{r}P^{r} K ⊆^{9.1.2(iii)} SP^{r} K ⊆^{9.1.1} SSPUp K =^{9.1.2(i)} SPUp K.

Second, Qeq(K) = Qeq(SP^{r} K) obviously holds (since quasiequations are preserved under taking subalgebras and reduced products). Therefore

ModQeq(K) = ModQeq(SP^{r} K) =^{9.1.3} SP^{r} K,

since, by Claim 9.1.2, SP^{r} K is closed under S and P^{r}. Therefore by (+) above, the proof of Theorem 9.1 is completed.

DEFINITION 9.1 (discriminator term):

(i) A class K of algebras is said to have a discriminator term iff there is a term τ(x, y, z, u) in the language of K such that in every member of K we have

τ(x, y, z, u) = \{ \begin{matrix}z, if x = y, \\ u, otherwise. \end{matrix}

(ii) A variety V is called a discriminator variety if the class Sir(V) of subdirectly irreducible members of V has a discriminator term.

Exercises 9.1:

(1) Show that if K has a discriminator term then K consists of simple algebras^{83} (and possibly the one-element algebra).

(2) Assume that the Boolean operations −, ∧, 0, 1 are available in K and that they satisfy the Boolean axioms (i.e. every element of K is a Boolean algebra with some further operations). This property will be referred to as ‘K has a Boolean reduct’. Prove that K has a discriminator term iff there is a term c(x) in the language of K such that

c(x) = \{ \begin{matrix}0, if x = 0, \\ 1, otherwise \end{matrix}

in every member of K. (Hint: τ(x, y, z, u) = [c(x ⊕ y) ∧ u] ∨ [z ∧ −c(x ⊕ y)]. Here ⊕ denotes symmetric difference.)

(3) Check how much simplification one can achieve in the proof of Thm.9.2 below under assuming that K has a Boolean reduct (cf. item (2) above).

83An algebra is simple iff it has no nontrivial congruences (and is nontrivial).

THEOREM 9.2. Let K be a class of similar algebras. Assume that K has a discriminator term. Then

HSP K = SPUp K.

The proof we give here is a model theory oriented, more intuitive proof as opposed to the more “computational” proofs in the standard universal algebra books. The idea of giving here such a more intuitive proof was suggested to us by Johan van Benthem.

Claim 9.2.1: Assume K has a discriminator term. Then to every quasiequation q in the language of K there is an equation e(q) such that (i)–(ii) below hold:

(i) K ⊧ (e(q) ↔ q), (ii) HSP K ⊧ (e(q) → q).

Proof of Claim 9.2.1: The proof goes by induction on the “length” of the quasiequation q, i.e. on the number of the equations on the left hand side of the implication in q. If there is no such equation that is, q is an equation itself then the claim is obvious.

The induction step: Any quasiequation q of form

(σ_{1} = ϱ_{1} ∧ ⋯ ∧ σ_{n} = ϱ_{n}) → (σ_{0} = ϱ_{0})

has an equivalent form

(σ_{1} = ϱ_{1}) → [(σ_{2} = ϱ_{2} ∧ ⋯ ∧ σ_{n} = ϱ_{n}) → (σ_{0} = ϱ_{0})].

That is, q is equivalent to (σ_{1} = ϱ_{1} → q′) for some quasiequation q′ shorter than q. Therefore, by the induction hypothesis, there is an equation σ = ϱ such that

(++)  K ⊧ ((σ = ϱ) ↔ q′) and HSP K ⊧ ((σ = ϱ) → q′).

Now, let τ(x, y, z, u) be a discriminator term in K. We define e(q) = e(σ_{1} = ϱ_{1} → q′) to be τ(σ_{1}, ϱ_{1}, σ, ϱ) = ϱ.

For proving the “right to left” implication of (i), assume that K |̸= e(q). Then there are some 𝔄 ∈ K and ā ∈ ^{ω}A with 𝔄 ⊧ (τ(σ_{1}, ϱ_{1}, σ, ϱ) ≠ ϱ)[ā]. Since τ is a discriminator term in K, it implies that 𝔄 ⊧ (σ ≠ ϱ)[ā]. Thus, by (++) above, 𝔄 |̸= q′[ā]. But 𝔄 ⊧ (σ_{1} = ϱ_{1})[ā] also follows from 𝔄 ⊧ (τ(σ_{1}, ϱ_{1}, σ, ϱ) ≠ ϱ)[ā], thus 𝔄 |̸= q[ā], i.e. K |̸= q.

To prove the other direction of (i), it is enough to prove (ii): Assume that for some 𝔄 ∈ HSP K, ā ∈ ^{ω}A, 𝔄 |̸= q[ā], i.e. 𝔄 |̸= (σ_{1} = ϱ_{1} → q′)[ā]. Then, 𝔄 ⊧ (σ_{1} = ϱ_{1})[ā] and, by (++), 𝔄 ⊧ (σ ≠ ϱ)[ā]. But τ is a disrciminator term in K, therefore HSP K ⊧ τ(x, x, z, u) = z, since discriminator terms satisfy this equation and equations are preserved under taking H, S, and P.^{84} Therefore

𝔄 ⊧ (τ(σ_{1}, ϱ_{1}, σ, ϱ) = σ ≠ ϱ)[ā]

proving HSP K |̸= e(q).

Proof of Theorem 9.2: First we prove that HSP K ⊧ Qeq(K). Indeed, let q ∈ Qeq(K). Then, by Claim 9.2.1 (i), there is an equation e(q) with K ⊧ (e(q) ↔ q). Since equations are preserved under HSP, HSP K ⊧ e(q) also holds. Now by Claim 9.2.1 (ii), HSP K ⊧ q.

Now

HSP K ⊆ ModQeq(K) =^{Thm.9.1} SPUp K,

which proves one direction of Thm.9.2.

84It is the x ≠ y ⟹ τ(x, y, z, u) = u part of the “discriminator property” of τ which is \underline{not} preserved from K to HSP K.

The other direction, SPUp K ⊆ HSP K is obvious, since Up ⊆ HP, PH ⊆ HP, SH ⊆ HS, and PP = P (cf. e.g. Monk [M76] p.387).

COROLLARY 9.1: Assume K has a discriminator term. Then (i) K is contained in some discriminator variety. (ii) The subdirectly irreducible members of HSP K are exactly the subdirectly irreducibles of

SUp K.

Proof: (ii): Let 𝔄 be a subdirectly irreducible member of HSP K. By Theorem 9.1, 𝔄 ∈ SP(SUp K). Then 𝔄 is a subdirect product of algebras from SUp K. By irreducibility, then 𝔄 ∈ SUp K. This proves (ii).

(i): The discriminator term τ which works for K also works for SUp K, since the discriminator property

∀x, y, z, u ([x ≠ y ⇒ τ(x, y, z, u) = u] ∧ [x = y ⇒ τ(x, y, z, u) = z])

is defined by a universal formula, thus is preserved under SUp. Thus SUp K has a discriminator term. But by (ii) the class Sir(HSP K) of subdirectly irreducibles of HSP K is in SUp K. Then by definition, HSP K is a discriminator variety.

Acknowledgement: During the years of the “evolution” of this work I received substantial help from so many colleagues (e.g. Johan van Benthem, Yde Venema, Peter Jipsen to mention a few) that I cannot list their names here; but I am grateful to them all. Special acknowledgement is due to Ildiko Sain whose input for this paper was extensive and valuable and I appreciate it much more than I can ever express.

68 LIST OF SYMBOLS

List of symbols

\underline{Classes of algebras} (the index n or α like in Cs_{n} or Cs_{α} referring to ranks of relations is omitted):

Symbol  Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . page:
BA  Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
BAO  Boolean Algebras with Operators . . . . . . . . . . . . . . . . . . . . . . 5,56
BRA^{0}  finitary approximation of BRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
BRA  Algebras of Binary Relations (without identity) . . . . . . . . . . . 4
BSR  Boolean semigroups of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
CA  Cylindric Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Crs  Cylindric relativized set algebras . . . . . . . . . . . . . . . . . . . . . . . . . 33
Cs  = (\begin{matrix}Cylindric set algebras \\ (Cs_{n}=algebras of n-ary relations) \end{matrix}) . . . . . . . . . . . . . . . . . . . . . . . . . 18
FSA  Finite Sequence Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
full Cs, full Crs, full Ps, etc.\}  = \binom{full (or powerset) elements of Cs etc.}{i.e. algebras with universe P(V) for some V} . . . . . . . .4,18,24,39,40
GFSA  Generalized Finite Sequence Algebras . . . . . . . . . . . . . . . . . . . . 51
Gs Generalized cylindric set algebra . . . . . . . . . . . . . . . . 24 and [HMTII]
HCA  Hyper Cylindric algebras, CA_{α} ⊂ HCA_{α} ⊂ RCA_{α} . . . . . . . . . . 21
L  = \binom{(Cs_{ω}^{reg}∩Lf_{ω})=lim_{n→ω} Cs_{n} i.e.}{class of algebras of finitary relations} . . . . . . . . . . . . . . . . . . . . . . . . 23
Lf  Locally finite dimensional cylindric algebras . . . . . . . . . .[HMT]
Lp  same as L but for the quasi-polyadic case . . . . . . . . . . . . . . . . . 42
Mod(Σ)  = \binom{class of all models}{of the set Σ of formulas} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
MsCA  Many sorted CA’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
NA  Nonassociative relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 61
PA  Polyadic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21,40,61 and [HMTII]
PaCA  Partial CA’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
PEA Polyadic Equality Algebras . . . . . . . . . . . . . . . . . . . . . 61 and [HMTII]
PK  class of Products of (members of) K . . . . . . . . . . . . . . . . . . . . . . . 4
Ps  Polyadic set algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
QPA  Quasi-Polyadic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41, 61
QPEA  Quasi-Polyadic Equality Algebras . . . . . . . . . . . . . . . . . . . . . . . . 61
Qps  Quasi polyadic set algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
QRA  Relation algebras with quasi-projective elements . . . . . [TG87]
RA Relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 and [HMTII]
RCA  Representable Cylindric Algebras . . . . . . . . . . . . . . . . . . . . . . . . 18
RdK  class of Reducts of (members of) K . . . . . . . . . . . . . . . . . . . . . . . 10
RPA  Representable Polyadic Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 39
RPEA  Representable Polyadic Equality Algebras . . . . . . . . . . . . . . . . 39
RQA  Representable Quasi-polyadic Algebras . . . . . . . . . . . . . . . . . . . 41
RQEA  Representable Quasi-polyadic Equality Algebras . . . . . . . . . . 42
RQPA  =RQA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
RRA  Representable Relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 13
RSC  Representable Substitution Cylindric algebras . . . . . . . . . . . . 43
RV  Representable members of the Variety V . . . . . . . . . . . . . . 20,61
SA  Semi-associative relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . 61
SC  Substitution Cylindric algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
SK  class of Subalgebras of (members of) K . . . . . . . . . . . . . . . . . . . . 4
TB_{S}  S-Transformational Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
IK  class of isomorphic copies of algebras in K . . . . . . . . . . . . . . . . 41
WA  Weakly associative relation Algebras . . . . . . . . . . . . . . . . . . . . . 61

LIST OF SYMBOLS 69

Algebras and their universes:

Symbol Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . page:
𝔉𝔫  algebra of Finitary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Frl  set of all Finitary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Gfs  set of Generalized finite sequences . . . . . . . . . . . . . . . . . . . . . . . . 51
Nr_{n}(𝔅) Neat reduct of 𝔅 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
𝔓𝔰𝔠  substitution cylindric Powerset algebra (a reduct of ℜ𝔢𝔩) . . 43
P(V) Powerset of V (= universe of 𝔓(V)) . . . . . . . . . . . . . . . . . . . . . . . 4
𝔓(V) = \binom{=⟨P(V),∩,∪,∅,V,−⟩}{Boolean algebra of subsets of V} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Ref  set of essentially finitary Relations . . . . . . . . . . . . . . . . . . . . . . . 37
ℜ𝔢𝔣  algebra of essentially finitary Relations . . . . . . . . . . . . . . . . . . . 37
ℜ𝔢𝔩_{α}  algebra of α-ary Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Rf  set of finitary Relations (universe of ℜ𝔣) . . . . . . . . . . . . . . . . . . 24
ℜ𝔣  algebra of finitary Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
ℜ𝔣𝔭  quasi-polyadic version of ℜ𝔣 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
𝔗𝔰  Two sorted cylindric-like algebra . . . . . . . . . . . . . . . . . . . . . . . . . 52

\underline{Other:}

Symbol Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . page:
c_{i}  \left\{{i-th cylindrification \atop (forgetting the i-th argument)}\right\} . . . . . . . . . . . . . . . . . . . 18
C_{Γ}  generalization of c_{i} . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Id  identity relation (both binary and n-ary) … 13,17
Id_{ij}  identity of i-th and j-th arguments . . . . . . . . . . . 24
p_{ij}  permutation of i and j . . . . . . . . . . . . . . . . . . . . . . . 39
s_{j}^{i}  substitution of j for i . . . . . . . . . . . . . . . . . . . . . . . . . 39
S_{τ}  substitution along τ . . . . . . . . . . . . . . . . . . . . . . . . . . 33
ω  set of all natural numbers . . . . . . . . . . . . . . . . . . . . 21
R ∘ S = {⟨a, b⟩ : ∃c(aRc and cSb)} . . . . . . . . . . . . . . . . . . . .4
nU  = U × U × ⋯ × U (n times), or equivalently . . 17
HU, ^{α}U ^{H}U is the set of all functions from H into U . . . . .
<ωU  set of all finite sequences over U . . . . . . . . . . . . . . . .4
|U|  the cardinality of U . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Keywords. Algebraic logic, predicate logics, cylindric algebras, relation algebras, polyadic algebras, quantifier logics, model theoretic logics, algebraic model theory, algebras of general relations, first order algebraic logic, abstract model theory, finitizability 1980 Mathematics subject classifications: 03G15, 03G35, 03G99, 03B99, 03C95, 03C99, 06F05, 08C10, 08C99

István Németi Mathematical Institute of the Hungarian Academy of Sciences Budapest, PF.127 H-1364, Hungary