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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