Lee monoid $L_4^1$ is non-finitely based

TL;DR

This paper introduces a new criterion to determine when monoids are non-finitely based and applies it to resolve the long-standing case of the Lee monoid $L_4^1$, establishing its non-finite basis.

Contribution

The paper provides a novel sufficient condition for non-finite basis in monoids and applies it to prove that the Lee monoid $L_4^1$ is non-finitely based, completing the finite basis problem for Lee monoids.

Findings

$L_4^1$ is non-finitely based.

New syntactic condition equivalent to Lee's sufficient condition.

Alternative proof that $L_ extell$ is non-finitely based for all $ extell \,\geq 3$.

Abstract

We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid $L_4^1$ is non-finitely based. The monoid $L_4^1$ was the only unsolved case in the finite basis problem for 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 also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new proof to the results of Zhang-Luo and Lee that the semigroup $L_\ell$ is non-finitely based each $\ell \ge 3$.