On the Minimal Degree of Rhodes Semisimple Semigroups

A finite semigroup is Rhodes semisimple if it admits a faithful completely reducible representation over the field of complex numbers. Rhodes determined these in the 1960s. We compute the minimal degree of a faithful representation of a Rhodes semisimple semigroup by partial and total transformations. This includes the case of 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. To illustrate what happens when a finite semigroup is not Rhodes semisimple, we show that the degree of the minimal faithful representation by total functions of the opposite of the full transformation monoid on n elements is 2^n.

This is joint work with Benjamin Steinberg.