  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.
