Cross-connection structure of concordant semigroups

TL;DR

This paper develops a categorical framework for understanding concordant semigroups, generalizing existing theories for regular semigroups through cross-connection structures and establishing a category equivalence.

Contribution

It characterizes categories from Green relations in concordant semigroups as consistent categories and proves their categorical equivalence, extending Nambooripad's analysis.

Findings

Categories from Green relations are consistent categories

Established a category equivalence for concordant semigroups

Generalized cross-connection analysis to broader semigroup classes

Abstract

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalised Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalisation of Nambooripad's cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.