Disjoint-union partial algebras

TL;DR

This paper characterizes classes of disjoint-union partial algebras of sets and functions, providing recursive axiomatizations, proving non-finite axiomatizability in general, and identifying conditions for finite axiomatization when intersection is included.

Contribution

It offers the first-order axiomatization of disjoint-union partial algebras, explores their properties under various signatures, and establishes finite axiomatizability with intersection.

Findings

Classes are recursively axiomatizable but not finitely axiomatizable without intersection.

Adding intersection makes classes finitely axiomatizable.

Certain classes are not closed under elementary equivalence.

Abstract

Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjoint-union partial algebra of sets but prove that no finite axiomatisation exists. We do the same for other signatures including one or both of disjoint union and subset complement, another partial binary operation we define. Domain-disjoint union is a partial binary operation on partial functions, returning the union if the arguments have disjoint domains and undefined otherwise. For each signature including one or both of domain-disjoint union and subset complement and optionally including composition, we consider the class of partial algebras isomorphic to a collection of partial functions closed under the operations. Again the classes prove to be axiomatisable, but not finitely axiomatisable, in first-order logic. We define the notion of pairwise combinability. For each of the previously considered signatures, we examine the class isomorphic to a partial algebra of sets/partial functions under an isomorphism mapping arbitrary suprema of pairwise combinable sets to the corresponding disjoint unions. We prove that for each case the class is not closed under elementary equivalence. However, when intersection is added to any of the signatures considered, the isomorphism class of the partial algebras of sets is finitely axiomatisable and in each case we give such an axiomatisation.