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Á. Kurucz and I. Németi:

#
Representability of Pairing Relation Algebras
depends on your ontology

This paper is about pairing relation algebras as well as fork algebras
and related subjects. In the 1991-92 fork algebra papers it was
conjectured that fork algebras admit a strong representation theorem
(w.r.t. ``real'' pairing). Then, this conjecture was disproved in the
following sense: a strong representation theorem for all abstract fork
algebras was proved to be impossible in most set theories including the
usual (well-founded) one as well as most non-well-founded set theories.
Here we show that the above quoted conjecture can still be made true by
choosing an appropriate set theory as our foundation of mathematics. (To
be precise, in its original form, the conjecture fails in every set
theory, but a natural, slight modification makes it true in our set
theory.) Namely, we show that there are non-well-founded set theories
in which every abstract fork algebra is representable in the strong sense,
i.e. it is isomorphic to a set relation algebra having a fork operation
which is obtained with the help of the real (set theoretic) pairing
function. Further, these non-well-founded set theories are consistent
if usual (ZF) set theory is consistent.

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