LEE MONOIDS ARE NONFINITELY BASED WHILE THE SETS OF THEIR ISOTERMS ARE FINITELY BASED

We establish a new sufficient condition under which a monoid is nonfinitely 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 $b$ with the relation $0=abab\cdots \,$ (length $\ell$ ). We show that every monoid $M$ which generates a variety containing $L_{5}^{1}$ and is contained in the variety generated by $L_{\ell }^{1}$ for some $\ell \geq 5$ is nonfinitely based. We establish this result by analysing $\unicode[STIX]{x1D70F}$ -terms for $M$ , where $\unicode[STIX]{x1D70F}$ is a certain nontrivial congruence on the free semigroup. We also show that if $\unicode[STIX]{x1D70F}$ is the trivial congruence on the free semigroup and $\ell \leq 5$ , then the $\unicode[STIX]{x1D70F}$ -terms (isoterms) for $L_{\ell }^{1}$ carry no information about the nonfinite basis property of $L_{\ell }^{1}$ .