On Some Finitary Conditions Arising from the Axiomatisability of Certain Classes of Monoid Acts

Abstract

This article considers those monoids S satisfying one or both of the finitary properties (R) and (r), focussing for the most part on inverse monoids. These properties arise from questions of axiomatisability of classes of S-acts, and appear to be of interest in their own right. If S is weakly right noetherian (WRN), that is, S has the ascending chain condition on right ideals, then certainly (r) holds. Other than this, we show that (R), (r), and (WRN) are independent. Our most detailed results are for Clifford monoids, in which case we completely characterise those S with trivial structure homomorphisms satisfying (R) or (r).

Key Words:

• Axiomatisable class
• Finitary conditions
• Inverse monoid
• S-act

2010 Mathematics Subject Classification:

• Primary: 20M10, 20M30

ACKNOWLEDGMENT

The authors acknowledge the support of EPSRC grant no. EP/I032312/1. Research also partially supported by the Hungarian Scientific Research Fund (OTKA) grant no. K83219. Some of this work appeared in the thesis of the third author, funded by HEC of Pakistan. The third author would like to thank the Department of Mathematics for hosting her as a Visiting Researcher in Spring 2012 when this work was completed.

Notes

Communicated by M. Kambites.