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Commutativity of operators on the lattice of existence varieties

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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 ρ P :Uρ P V⇔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 α P :Uα P V⇔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 [U ρ,U ρ] so that there is associated with each such congruence an idempotent operatorU→U ρ on ℒev(ℛI). This paper establishes numerous results concerning the commutativity of operators of this form.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Simon Fraser University, V5A 1S6, Burnaby, B.C., Canada

    Norman R. Reilly & Shuhua Zhang

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  1. Norman R. Reilly
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  2. Shuhua Zhang
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This work was supported in part by NSERC Grant 4044.

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Reilly, N.R., Zhang, S. Commutativity of operators on the lattice of existence varieties. Monatshefte für Mathematik 123, 337–364 (1997). https://doi.org/10.1007/BF01326768

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