Domain Range Semigroups and Finite Representations

TL;DR

This paper investigates the finite representability of domain-range semigroups and related algebraic structures, proving non-finite axiomatisability in some cases and establishing finite representation properties for others.

Contribution

It demonstrates that certain classes of domain-range semigroups are not finitely axiomatisable and shows that some relation algebra reducts have the finite representation property.

Findings

Representation class of domain-range semigroups with demonic composition is not finitely axiomatisable.

Ordered domain algebras with specific signatures have the finite representation property.

Survey of results related to the finite representation property in algebraic logic.

Abstract

Relational semigroups with domain and range are a useful tool for modelling nondeterministic programs. We prove that the representation class of domain-range semigroups with demonic composition is not finitely axiomatisable. We extend the result for ordered domain algebras and show that any relation algebra reduct signature containing domain, range, converse, and composition, but no negation, meet, nor join has the finite representation property. That is any finite representable structure of such a signature is representable over a finite base. We survey the results in the area of the finite representation property.