Reducts of Relation Algebras - The Aspects of Axiomatisability and Finite Representability.

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 isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson. We also show that the class of representable join semilattice-ordered semigroups has a recursively enumerable axiomatisation using back-and-forth games.