Model-theoretic properties of free, projective, and flat S-acts

TL;DR

This paper surveys the model-theoretic properties of free, projective, and flat S-acts over monoids, focusing on axiomatizability, completeness, and stability, with some new results on axiomatizability conditions.

Contribution

It provides a comprehensive review of the model theory of these classes of S-acts and introduces new criteria for axiomatizability of free S-acts.

Findings

Characterization of monoids with axiomatizable free S-acts

Analysis of stability and model completeness for classes of S-acts

Most results previously published, with some new axiomatizability conditions

Abstract

This is the second in a series of articles surveying the body of work on the model theory of S-acts over a monoid S. The first concentrated on the theory of regular S-acts. Here we review the material on model-theoretic properties of free, projective, and (strongly, weakly) flat S-acts. We consider questions of axiomatizability, completeness, model completeness, and stability for these classes. Most but not all of the results have already appeared; we remark that the description of those monoids S such that the class of free left S-acts is axiomatizable, is new.