Generators and relations of direct products of semigroups

Abstract

The purpose of this paper is to give necessary and sufficient conditions for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequence we construct a semigroup $S$ of order 11 such that $S\times T$ is finitely generated but not finitely presented for every finitely generated infinite semigroup $T$. By way of contrast we show that, if $S$ and $T$ belong to a wide class of semigroups, then $S\times T$ is finitely presented if and only if both $S$ and $T$ are finitely presented, exactly as in the case of groups and monoids.