We give an algebraic version of first order logic without equality in which the class of representable algebras forms a finitely based equational class. Further, the representables are defined in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1 (a concrete version of which is given by Theorems 1.8 and 3.2), while its logical form is Corollary 4.2.
We give an algebraic version of first order logic with equality as well, but only in a weaker form. See section 5.1, especially Remark 5.5 there.
The proof of Theorem 1 is elaborated in sections 2 and 3. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 3.2 is a purely semigroup theoretic result. Cf. also "Further main results" in the Introduction.