Some more axiomatisability for S-acts

TL;DR

This paper investigates conditions under which classes of flat S-acts are axiomatisable, providing a general procedure and revealing multiple methods for some classes, advancing the understanding of S-act axiomatisability.

Contribution

It introduces a general procedure for axiomatizing flatness classes of S-acts and uncovers multiple axiomatization methods for certain classes, a novel finding.

Findings

Identified necessary and sufficient conditions for axiomatisability.

Developed a general procedure for axiomatizing flat S-acts.

Discovered multiple axiomatization methods for some classes.

Abstract

This paper discusses necessary and sufficient conditions on a monoid S, such that the class of C-flat left $S$-acts is axiomatisable, where C is the class of all embeddings (of right ideals into S) of right S-acts. We consider the axiomatisability of some flatness classes of S-acts, which were previously discussed by Bulman-Fleming and Gould [1] . We present here a more general procedure to axiomatise these classes. A similar type of general results have been found for S-posets by Gould and Shaheen [10]. We have found that there are some classes of S-acts which are axiomatisable by more than one method. This has not been seen before.