Generators and presentations for direct and wreath products of monoid acts

Abstract

We investigate the preservation of the properties of being finitely generated and finitely presented under both direct and wreath products of monoid acts. A monoid M is said to preserve property {mathcal {P}} in direct products if, for any two M-acts A and B, the direct product Atimes B has property {mathcal {P}} if and only if both A and B have property {mathcal {P}}. It is proved that the monoids M that preserve finite generation (resp. finitely presentability) in direct products are precisely those for which the diagonal M-act Mtimes M is finitely generated (resp. finitely presented). We show that a wreath product Awr B is finitely generated if and only if both A and B are finitely generated. It is also proved that a necessary condition for Awr B to be finitely presented is that both A and B are finitely presented. Finally, we find some sufficient conditions for a wreath product to be finitely presented.