Some results on relation algebra reducts: Residuated and semilattice-ordered semigroups

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 Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called R-models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups.