An introduction to presentations of monoid acts: quotients and subacts

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.