Inductive groupoids and cross-connections of regular semigroups

TL;DR

This paper demonstrates a direct equivalence between two major structural representations of regular semigroups—inductive groupoids and cross-connections—showing how to construct one from the other.

Contribution

It establishes a direct equivalence between inductive groupoid and cross-connection constructions for regular semigroups, unifying two major approaches.

Findings

Equivalence between inductive groupoid and cross-connection representations.

Construction methods for deriving one structure from the other.

Unification of two major regular semigroup theories.

Abstract

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann--Schein--Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.