Commutativity of operators on the lattice of existence varieties

Abstract

Abstract. In a previous paper the authors introduced seven complete congruences on the lattice L, ee~(.r ) of e-varieties of regular semigroups of the form pe:qlpe~K ",r ~ o q/= ~ o ~U, where ~ is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form c~: q/c~e~ ~.~ c~ q/= ~ c~ ~, where ~ is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence p on Lf~v(~5 ~ and each q/e ~o,(~5~), the p-class of q/is an interval [q/p, q/P] so that there is associated with each such congruence an idempotent operator q/~ q/P on 5rThis paper establishes numerous results concerning the commutativity of operators of this form.

Operators on the Lattice of Pseudovarieties of Finite Semigroups

“…In particular, the present authors have extended this approach to the lattice of existence varieties [28] and [29] while Auinger, Hall and the present authors extended these techniques to the lattice of pseudovarieties [7].…”