A sufficient condition under which a semigroup is nonfinitely based. (Q2814345)

A variety of algebras is a limit variety if it is minimal with respect to being non-finitely based. 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 \(\mathbf{J_2}\). It can be inferred from the work of the first author [Semigroup Forum 86, No. 1, 212-220 (2013; Zbl 1273.20062)] that further examples must exist.NEWLINENEWLINE The goal of the paper under review is to show that a specific monoid -- the product of a certain seven-element aperiodic monoid \(A^1\) with its dual -- is not finitely based. In a sequel [``A new example of a limit variety of aperiodic monoids''], the variety it generates will be shown to be a limit variety. The nonfinite basis property is achieved by application of the result stated in the title. The condition states, for any given integer \(n\geq 2\), that a certain infinite set of identities should be satisfied, but each member of a further finite set of identities must not be satisfied.