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 Band 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. Weshow 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.