A note on the definition of small overlap monoids

TL;DR

This paper clarifies and extends the definition of small overlap monoids, ensuring the author's recent methods apply to the original conditions introduced by Remmers, thereby broadening their theoretical framework.

Contribution

It removes the discrepancy in the definition of small overlap conditions, generalizing the author's previous results to fully encompass Remmers' original monoids.

Findings

Unified the definition of small overlap monoids with Remmers' original concept.

Extended existing methods to apply to the full generality of Remmers' conditions.

Ensured broader applicability of small overlap monoid theory.

Abstract

Small overlap conditions are simple and natural combinatorial conditions on semigroup and monoid presentations, which serve to limit the complexity of derivation sequences between equivalent words in the generators. They were introduced by J.H.Remmers, and more recently have been extensively studied by the present author. However, the definition of small overlap conditions hitherto used by the author was slightly more restrictive than that introduced by Remmers; this note eliminates this discrepancy by extending the recent methods and results of the author to apply to Remmers' small overlap monoids in full generality.