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

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