Structure of Clifford Semigroups of Matrices

TL;DR

This paper characterizes the structure of Clifford semigroups of matrices over any field, showing they are isomorphic to subdirect products of linear groups, and extends the result to infinite matrices.

Contribution

It provides a complete characterization of Clifford semigroups of matrices, including finite and countably infinite cases, as subdirect products of linear (0-)groups.

Findings

Finite order Clifford semigroups are subdirect products of linear (0-)groups.

The characterization extends to countably infinite matrices.

Necessary and sufficient conditions for Clifford semigroups of matrices.

Abstract

In this paper, we characterize completely the structure of Clifford semigroups of matrices over an arbitrary field. It is shown that a semigroups of matrices of finite order is a Clifford semigroup if and only if it is isomorphic to a subdirect product of some linear (0-)groups. Then we generalize this result to a semigroup of matrices of countably infinite order and give a similar necessary and sufficient condition for it to be a Clifford semigroup.