Lee monoid \(L_4^1\) is non-finitely based

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 \(L_\ell \) 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 for each \(\ell \ge 3\).