Self-dual varieties and networks in the lattice of varieties of completely regular semigroups

TL;DR

This paper investigates the structure of kernel classes in the lattice of varieties of completely regular semigroups, revealing that many contain multiple copies of the lattice of varieties of bands, especially among self-dual varieties.

Contribution

It develops the concept of duality in the lattice and analyzes the kernel classes of various subvarieties, including self-dual varieties, showing their complex internal structure.

Findings

Kernel classes of many varieties contain multiple band lattice copies.

Self-dual varieties often have kernel classes with rich sublattice structures.

The study extends understanding of the lattice structure of completely regular semigroups.

Abstract

The kernel relation $K$ on the lattice $\mathcal{L}(\mathcal{CR})$ of varieties of completely regular semigroups has been a central component in many investigations into the structure of $\mathcal{L}(\mathcal{CR})$. However, apart from the $K$-class of the trivial variety, which is just the lattice of varieties of bands, the detailed structure of kernel classes has remained a mystery until recently. Kad'ourek [RK2] has shown that for two large classes of subvarieties of $\mathcal{CR}$ their kernel classes are singletons. Elsewhere (see [RK1], [RK2], [RK3]) we have provided a detailed analysis of the kernel classes of varieties of abelian groups. Here we study more general kernel classes. We begin with a careful development of the concept of duality in the lattice of varieties of completely regular semigroups and then show that the kernel classes of many varieties, including many self-dual varieties, of completely regular semigroups contain multiple copies of the lattice of varieties of bands as sublattices.