Generators and presentations for direct and wreath products of monoid acts
Craig J. Miller · Semigroup Forum · 2018
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 \(A\times B\) 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 \(M\times M\) is finitely generated (resp. finitely presented). We show that a wreath product \(A\wr B\) is finitely generated if and only if both A and B are finitely generated. It is also proved that a necessary condition for \(A\wr B\) 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.