Finiteness conditions of wreath products of semigroups and related properties of diagonal acts

Abstract

The purpose of this thesis is to consider finite generation, finite presentability and related properties of restricted wreath products of semigroups. We show that the wreath product Awr B of two monoids is finitely generated if and only if A and B are finitely generated and the action by right multiplication on B of the group of units of B has only finitely many orbits. Also we show that the wreath product AwrB of two non-trivial monoids is finitely presented if and only if A is finitely presented and B is finite. The situation is more complicated in the case of the wreath product SₑwrT of two semigroups with respect to an idempotent e ϵ S. We give a complete characterization for finite generation in the case where T is finite. This result depends on the properties of the diagonal action of S on S x S. We also prove that if this action is not finitely generated, then SₑwrT (with S infinite and T finite) is finitely presented if and only if S x S is finitely presented and T is the direct product of a monoid and a left zero semigroup. In the case where T is infinite, we prove that S must be a monoid in order for SwrT to be finitely generated. We show that the finiteness properties of periodicity and local finiteness are preserved under the wreath product construction. We conclude the thesis with a systematic investigation into the properties of diagonal acts of semigroups, and make some interesting connections between diagonal acts and power semigroups.