On the minimal faithful degree of Rhodes semisimple semigroups

In this paper we compute the minimal degree of a faithful rep-resentation by partial transformations of a finite semigroup admitting a faithful completely reducible matrix represen-tation 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 bi-nary relations (i.e., matrices over the Boolean semiring). Our answer reduces the computation to considerations of permu-tation 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 minimal faithful de-gree of the opposite monoid of the full transformation monoid.& COPY; 2023 Elsevier Inc. All rights reserved.