Existence of a new limit variety of aperiodic monoids (Q1941751)

A variety of algebras is a `limit' variety if it is minimal with respect to being non-finitely based. As the author points out, since it appears to be hopeless to describe all limit varieties of groups it is reasonable, in the context of varieties of monoids, to focus on the aperiodic varieties: those consisting of monoids with no nontrivial subgroups. \textit{M. Jackson} [Semigroup Forum 70, No. 2, 159-187 (2005; Zbl 1073.20052)] constructed two concrete examples \(\mathbf J_1\) and \(\mathbf J_2\), generated respectively by finite monoids \(J_1\) and \(J_2\). Here the author answers a question posed therein by showing that the monoid \(L^1\), obtained by adjoining an identity element to the six-element semigroup \(L=\langle a,b\mid a^2=a,\;b^2=b,\;aba=0\rangle\), is non-finitely based and satisfies identities not satisfied in \(\mathbf J_1\) or in \(\mathbf J_2\). While the author does not know if the variety generated by \(L^1\) is itself a limit variety, by Zorn's Lemma it must contain such a variety, and that variety is different from \(\mathbf J_1\) and \(\mathbf J_2\).