Lee monoids are non-finitely based while the sets of their isoterms are finitely based

TL;DR

This paper introduces a new criterion to determine when certain monoids, specifically Lee monoids, are non-finitely based, and demonstrates that their sets of isoterms are finitely based, revealing nuanced structural properties.

Contribution

It provides a novel sufficient condition for non-finite basis of monoids and applies it to Lee monoids, analyzing $ au$-terms to distinguish between finite and infinite bases.

Findings

Lee monoids $L_ ell^1$ are non-finitely based for $ ell \uge 5$

Sets of isoterms for $L_ ell^1$ are finitely based when $ ell \ule 5$

Analysis of $ au$-terms clarifies the basis properties of Lee monoids

Abstract

We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We show that every monoid which generates a variety containing $L_5^1$ and is contained in the variety generated by $L_\ell^1$ for some $\ell \ge 5$ is non-finitely based. We establish this result by analyzing $\tau$-terms for $M$ where $\tau$ is certain non-trivial congruence on the free semigroup, that is, we analyze words $\bf u$ with the property that ${\bf u} \tau {\bf v}$ whenever $M$ satisfies an identity ${\bf u} \approx {\bf v}$. We also show that if $\tau$ is the trivial congruence on the free semigroup and $\ell \le 5$ then the $\tau$-terms (isoterms) for $L_\ell^1$ carry no information about the non-finite basis property of $L_\ell^1$.