Reducts of relation algebras: The aspects of axiomatisability and finite representability

TL;DR

This paper investigates the axiomatisability and finite representability of certain algebraic structures, proving finite representability for residuated semigroups and establishing axiomatization properties for join semilattice-ordered semigroups.

Contribution

It proves finite representability for all finite residuated semigroups and introduces representability games for join semilattice-ordered semigroups, advancing understanding of their axiomatization.

Findings

Finite representability property for residuated semigroups.

Recursively enumerable axiomatisation for join semilattice-ordered semigroups.

Introduction of representability games for algebraic structures.

Abstract

In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatisation. For this purpose, we introduce representability games for join semilattice-ordered semigroups.