2018/04/06 by Miller, Craig
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1804.03010
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× 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× 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.