An introduction to presentations of monoid acts: quotients and subacts

TL;DR

This paper develops a comprehensive theory for presentations of monoid acts, focusing on subacts, quotients, and unions, and establishes conditions for finite presentability in these contexts.

Contribution

It introduces methods to construct presentations for subacts, quotients, and unions of monoid acts, and provides finite presentability results under various conditions.

Findings

Constructed presentations for subacts and quotients from a given act.

Derived an act's presentation from those of a subact and a quotient.

Established finite presentability conditions for acts based on monoid properties.

Abstract

The purpose of this paper is to introduce the theory of presentations of monoids acts. We aim to construct `nice' general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an $M$-act $A$ and a subact $B$ of $A$, on the one hand we construct presentations for $B$ and the Rees quotient $A/B$ using a presentation for $A$, and on the other hand we derive a presentation for $A$ from presentations for $B$ and $A/B$. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact $B$ has finite complement in an $M$-act $A$. We show that if $M$ is a finitely generated monoid and $B$ is finitely presented, then $A$ is finitely presented. We also show that if $M$ belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds.