On finite generation and other finiteness conditions for wreath products of semigroups

The purpose of the paper is to investigate finite generation and some other basic finiteness conditions for (restricted) wreath products of semigroups (with respect to an idempotent e). The main result is that such a wreath product S(e)wrT, with T finite, is finitely generated if and only if S-2 = S, T-2 = T, S is finitely generated and either S x S is a finitely gene-rated S-act, or else every element of T belongs to the principal right ideal of a right identity. Further results are obtained for the case where T is infinite, and also for finite presentability, periodicity and local finiteness.