A semilattice of varieties of completely regular semigroups

Abstract

Completely regular semigroups are unions of their (maximal) subgroups with the unary operation within their maximal subgroups. As such they form a variety whose lattice of subvarieties is denoted by $\mathcal L(\mathcal C\mathcal R)$. We construct a 60-element $\cap$-subsemilattice and a 38-element sublattice of $\mathcal L(\mathcal C\mathcal R)$. The bulk of the paper consists in establishing the necessary joins for which it uses Polák's theorem.

Keywords

• completely regular semigroup
• lattice
• variety
• $\cap$-subsemilattice