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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