Commutativity of operators on the lattice of existence varieties

Abstract

In a previous paper the authors introduced seven complete congruences on the lattice ℒev(ℛI of e-varieties of regular semigroups of the form ρP:UρPV⇔P∘U=P∘V, whereP 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 αP:UαPV⇔P∩U=P∩V, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence ρ on ℒev(ℛI) and eachU∈ℒev(ℛI), the ρ-class ofU is an interval [Uρ,Uρ] so that there is associated with each such congruence an idempotent operatorU→Uρ on ℒev(ℛI). This paper establishes numerous results concerning the commutativity of operators of this form.