Commutativity of operators on the lattice of existence varieties (Q1357594)

A class of regular semigroups is called an e-variety if it is closed under direct products, homomorphic images and regular subsemigroups. The article continues earlier work of the authors [J. Algebra 178, No. 3, 733-759 (1995; Zbl 0842.20051); Semigroup Forum 53, No. 1, 1-24 (1996; Zbl 0854.20069)]. In the former article, the authors introduced 7 complete congruences on the lattice \({\mathbf L}\) of e-varieties of regular semigroups. In the work under review, 4 new complete congruences on \({\mathbf L}\) are introduced. They have the form \(\alpha_{\mathcal P}\): \(\mathcal A\alpha_{\mathcal P}\mathcal B\Leftrightarrow\mathcal A\cap\mathcal P=\mathcal B\cap\mathcal P\) where \(\mathcal P\) is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. To each complete congruence \(\rho\) on \({\mathbf L}\) is associated an idempotent operator \({\mathcal A}\rightarrow{\mathcal A}^{\rho}\) on \({\mathbf L}\). Numerous results concerning the commutativity of such operators are established.