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

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