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 L41 is non-finitely based. The monoid L41 was the only unsolved case in the finite basis problem for Lee monoids Lℓ1, obtained by adjoining an identity element to the semigroup Lℓ generated by two idempotents a and b subjected to the relation 0=abab⋯ (length ℓ). 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 Lℓ is non-finitely based for each ℓ≥3.