Ladders and canonical varieties of completely regular semigroups

Abstract

A completely regular semigroup is a (disjoint) union of its (maximal) subgroups. We consider it here with the unary operation of inversion within its maximal subgroups. Their totality $$\mathcal {C}\mathcal {R}$$ forms a variety whose lattice of subvarieties is denoted by $$\mathcal {L}(\mathcal {C}\mathcal {R})$$ . On it, one defines the relations $$\mathbf {B}^\wedge $$ and $$\mathbf {B}^\vee $$ by $$\begin{aligned} \begin{array}{lll} \mathcal {U}\ \mathbf {B}^\wedge \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\cap \mathcal {B} =\mathcal {V}\cap \mathcal {B}, \\ \mathcal {U}\ \mathbf {B}^\vee \ \mathcal {V}&{} \Longleftrightarrow &{} \mathcal {U}\vee \mathcal {B} =\mathcal {V}\vee \mathcal {B} , \end{array} \end{aligned}$$ respectively, where $$\mathcal {B}$$ denotes the variety of all bands. This is a study of the interplay between the $$\cap $$ -subsemilatice $$\triangle $$ of $$\mathcal {L}(\mathcal {C}\mathcal {R})$$ of upper ends of $$\mathbf {B}^\wedge $$ -classes and their $$\mathbf {B}^\vee $$ -classes. The main tool is the concept of a ladder and their $$\mathbf {B}^\vee $$ -classes, an indispensable part of the important Polak’s theorem providing a construction for the join of varieties of completely regular semigroups. The paper includes the tables of ladders of the upper ends of most $$\mathbf {B}^\wedge $$ -classes. Canonical varieties consist of two ascending countably infinite chains which generate most of the upper ends of $$\mathbf {B}^\wedge $$ -classes.