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In this paper we compute the minimum degree of a faithful representation bypartial transformations of a finite semigroup admitting a faithful completelyreducible matrix representation over the field of complex numbers. Thisincludes all inverse semigroups, and hence our results generalize earlierresults of Easdown and Schein on the minimal faithful degree of an inversesemigroup. It also includes well-studied monoids like full matrix monoids overfinite fields and the monoid of binary relations (i.e., matrices over theBoolean semiring). Our answer reduces the computation to considerations ofpermutation representations of maximal subgroups that are faithful whenrestricted to distinguished normal subgroups. This is analogous to (andinspired by) recent results of the second author on the minimal number ofirreducible constituents in a faithful completely reducible complex matrixrepresentation of a finite semigroup that admits one. To illustrate whathappens when a finite semigroup does not admit a faithful completely reduciblerepresentation, we compute the minimum faithful degree of the opposite monoidof the full transformation monoid.

On the minimal faithful degree of Rhodes semisimple semigroups