Ildikó Sain and István Németi:

Fork Algebras in Usual and in Non-well-founded Set Theories (An Overview)

We consider classes of relation algebras expanded with new operations based on the formation of ordered pairs. Examples for such algebras are fork algebras of computer science and pairing algebras of algebraic logic. Among others we prove that the equational theory of the class of Set (or Proper) Fork Algebras (where the fork operation is defined with the help of standard set theoretic pairing) is not finitely axiomatizable, not only in our usual set theory, but also in most of the known unusual set theories, e.g. in the non-well-founded ones surveyed in Aczel's book, in Quine's New Foundation, etc. We also attempt to present a kind of ``roadmap'' by giving a systematic answer to the question asking which choice of proper fork (or pairing) algebras in which choice of set theory can support a reasonable representation theory. It turns out that the Axiom of Strong Extensionality plays a decisive role.
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