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 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 ≀ 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.2010 Mathematics Subject Classification. 20M30, 20M05.

On Presentations of Semigroup of Transformations Restricted by an Equivalence

“…This section is strongly motivated by the work of [38], [23], [1] and [39]. Using the generating set and presentation of E(X, σ), we construct the direct product A × B, which leads to the characterisations of the monoids M that have the property that, for any two M − acts A and B, the direct product A × B is finitely generated (resp.…”

Semigroups for which every right congruence of finite index is finitely generated

“…One may consult [12, Section 1.5] for further details. A more systematic study of finite presentability of monoid acts was developed in [16,17].…”