On Certain Cross Varieties of Aperiodic Monoids with Commuting Idempotents

Abstract

Let \({\fancyscript{A}_\mathrm{com}}\) denote the class of aperiodic monoids with commuting idempotents. It is shown that any subvariety of \({\fancyscript{A}_\mathrm{com}}\) that satisfies the identity \({x^2yx \approx xyx^2}\) is Cross if and only if it excludes the almost Cross varieties \({\mathbf{J_1}, \mathbf{J_2}}\), and \({\mathbf{L}}\). Consequently, these three almost Cross varieties are unique among all subvarieties of \({\fancyscript{A}_\mathrm{com}}\) that satisfy the identity \({x^2yx \approx xyx^2}\). On the other hand, the existence of almost Cross subvarieties of \({\fancyscript{A}_\mathrm{com}}\) different from \({\mathbf{J_1}, \mathbf{J_2}}\), and \({\mathbf{L}}\) is also established.