Cross-connections in Clifford semigroups

TL;DR

This paper explores the structural properties of Clifford semigroups through the lens of cross-connection theory, revealing isomorphisms and degeneracies in their structural descriptions.

Contribution

It demonstrates that the semigroup of normal cones is isomorphic to the original Clifford semigroup, even without monoid assumptions, and analyzes the cross-connection structure in semilattices.

Findings

Semigroup of normal cones is isomorphic to the original Clifford semigroup.

Cross-connection description degenerates in Clifford semigroups.

Provides a detailed description of cross-connection structures in semilattices.

Abstract

An inverse Clifford semigroup (often referred to as just a Clifford semigroup) is a semilattice of groups. It is an inverse semigroup and in fact, one of the earliest studied classes of semigroups. In this short note, we discuss various structural aspects of a Clifford semigroup from a cross-connection perspective. In particular, given a Clifford semigroup, we show that the semigroup of normal cones is isomorphic to the original semigroup, even when it is not a monoid. Hence, we see that cross-connection description degenerates in Clifford semigroups. Further, we specialise the discussion to provide the description of the cross-connection structure in an arbitrary semilattice, also.