Polák theorem and $\mathbf {B}$-relation on varieties of completely regular semigroups

Abstract

Completely regular semigroups $\mathcal {C}\mathcal {R}$ are unions of their subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by $\mathcal {L}(\mathcal {C}\mathcal {R})$. The Polák theorem concerns the computation of joins in $\mathcal {L}(\mathcal {C}\mathcal {R})$. The $\mathbf {B}$-relation on $\mathcal {L}(\mathcal {C}\mathcal {R})$ identifies varieties with the same bands. We elaborate upon two nontrivial conditions in Polák’s theorem applied to certain subsets of $\mathcal {C}\mathcal {R}$ which amounts to solving particular equations in $\mathcal {L}(\mathcal {C}\mathcal {R})$.