On the minimal faithful degree of Rhodes semisimple semigroups

TL;DR

This paper determines the minimal degree of faithful partial transformation representations for finite semigroups with completely reducible complex matrix representations, generalizing previous inverse semigroup results and applying to various monoids.

Contribution

It provides a formula for the minimal faithful degree of a broad class of semigroups, linking permutation representations of subgroups to faithful matrix representations.

Findings

Reduces the problem to permutation representations of maximal subgroups.

Generalizes earlier inverse semigroup results.

Includes applications to monoids like full matrix monoids and Boolean relations.

Abstract

In this paper we compute the minimum degree of a faithful representation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix representation over the field of complex numbers. This includes all inverse semigroups, and hence our results generalize earlier results of Easdown and Schein on the minimal faithful degree of an inverse semigroup. It also includes well-studied monoids like full matrix monoids over finite fields and the monoid of binary relations (i.e., matrices over the Boolean semiring). Our answer reduces the computation to considerations of permutation representations of maximal subgroups that are faithful when restricted to distinguished normal subgroups. This is analogous to (and inspired by) recent results of the second author on the minimal number of irreducible constituents in a faithful completely reducible complex matrix representation of a finite semigroup that admits one. To illustrate what happens when a finite semigroup does not admit a faithful completely reducible representation, we compute the minimum faithful degree of the opposite monoid of the full transformation monoid.