Generators and presentations for direct and wreath products of monoid acts

TL;DR

This paper studies how properties like finite generation and presentation are preserved under direct and wreath products of monoid acts, providing characterizations and conditions for these properties to hold.

Contribution

It characterizes monoids that preserve finite generation and presentation in direct products and establishes conditions for wreath products to be finitely generated or presented.

Findings

Monoids preserving finite generation are characterized by their diagonal act being finitely generated.

Wreath product is finitely generated if and only if both factors are finitely generated.

Necessary and sufficient conditions for wreath product to be finitely presented are provided.

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 $A\times B$ 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.