Generators and presentations for direct and wreath products of monoid acts
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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 P in direct products if, for any two M-acts A and B, the direct product A x B has property P if and only if both A and B have property 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 x M is finitely generated (resp. finitely presented). We show that a wreath product A ≀ 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 ≀ 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.
Miller , C 2018 , ' Generators and presentations for direct and wreath products of monoid acts ' , Semigroup Forum , vol. First Online . https://doi.org/10.1007/s00233-018-9987-5
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