Pseudovarieties of ordered completely regular semigroups

TL;DR

This paper explores the classification of ordered completely regular semigroups within pseudovarieties, revealing structural properties and their implications for semigroup theory and computer science applications.

Contribution

It introduces the role of order in pseudovarieties of completely regular semigroups and analyzes the lattice structure of these pseudovarieties.

Findings

The lattice of pseudovarieties of ordered completely regular semigroups is modular.

Intersection with the pseudovariety of bands defines a complete endomorphism.

The paper characterizes new pseudovarieties arising in the ordered context.

Abstract

This paper is a contribution to the theory of finite semigroups and their classification in pseudovarieties, which is motivated by its connections with computer science. The question addressed is what role can play the consideration of an order compatible with the semigroup operation. In the case of unions of groups, so-called completely regular semigroups, the problem of which new pseudovarieties appear in the ordered context is solved. As applications, it is shown that the lattice of pseudovarieties of ordered completely regular semigroups is modular and that taking the intersection with the pseudovariety of bands defines a complete endomorphism of the lattice of all pseudovarieties of ordered semigroups.