Fork Algebras in Usual and in Non-well-founded Set Theories1

Due to their high expressive power and applicability in computer science, fork algebras have intensively been studied lately. In particular, they have been fruitfully applied e.g. in the theory of programming (specification, semantics etc.). The literature of fork algebras has been alive and active for at least five years by now. Some references are: [34], [35], [18], [36], [10], [11], [14], [8], [9], [15], [16] and [17]. Analogously to the situation with Boolean algebras, groups, semigroups, polyadic algebras etc., it is considered desirable to develop a representation theory for fork algebras, too (cf. e.g. [14, [10]). This would consist of • defining a “concrete” class of, say, “set (or proper) fork algebras” (analogously to Boolean set algebras, transformation semigroups, polyadic set algebras), and • an axiomatic class FA, defined by finitely many equations (or perhaps quasi-equations), of “abstract fork algebras”; and then • a “representation” theorem stating that every member of FA is isomorphic to a set fork algebra. Here, among others, we look at various possible choices for the concrete class that could play the rôle of set or proper fork algebras in such a representation theorem. Some of the candidate classes for this rôle were investigated in [26], [29], [23]. In [26], [29] and [23], the following results were proved in usual set theory (ZFC). The equational theory Eq(TPA) of True Pairing Algebras (TPA’s) and the equational theory Eq(SFA) of Proper (or Set) Fork Algebras (SFA’s) are not recursively enumerable, moreover, they both are Π1hard. The Axiom of Foundation was mentioned in the proofs. Therefore,