Relations on a lattice of varieties of completely regular semigroups

Abstract

Completely regular semigroups $\mathcal{CR}$ are considered here with the unary operation of inversion within the maximal subgroups of the semigroup. This makes $\mathcal{CR}$ a variety; its lattice of subvarieties is denoted by $\mathcal{L(CR)}$. We study here the relations ${\mathbf K,T,L}$ and ${\mathbf C}$ relative to a sublattice $\Psi$ of $\mathcal{L(CR)}$ constructed in a previous publication. For ${\mathbf R}$ being any of these relations, we determine the ${\mathbf R}$-classes of all varieties in the lattice $\Psi$ as well as the restrictions of ${\mathbf R}$ to $\Psi$.

Keywords

• semigroup
• completely regular
• variety
• lattice
• relation
• kernel
• trace
• local relation
• core