Polák theorem and $$\mathbf {B}$$ 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 Polak 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 Polak’s theorem applied to certain subsets of \(\mathcal {C}\mathcal {R}\) which amounts to solving particular equations in \(\mathcal {L}(\mathcal {C}\mathcal {R})\).