2017/09/26 by Miller, Craig, Ruskuc, Nik
#20M05 #20M30 #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1709.08916
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.