Cross-connections of completely simple semigroups

TL;DR

This paper explores the structure of completely simple semigroups through the lens of normal categories and cross-connections, providing a new representation of these semigroups as cross-connection semigroups.

Contribution

It identifies the normal categories associated with completely simple semigroups and characterizes their cross-connections, linking sandwich matrices to these structures.

Findings

Normal categories for completely simple semigroups are identified.

The semigroup of normal cones is shown to be a semi-direct product.

Complete semigroups are represented as cross-connection semigroups.

Abstract

A completely simple semigroup S is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that S is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup. In the study of structure theory of regular semigroups, K.S.S. Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left(right) ideals using cross-connections. A normal category C is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories C and D is a local isomorphism frm D to N*C where N*C is the normal dual of the category C. In this paper, we identify the normal categories associated with a completely simple semigroup S and show that the semigroup of normal cones TL(S) is isomorphic to a semi-direct product of semigroups. We characterize the cross-connections in this case and show that each sandwich matrix P correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.