On the minimal faithful degree of Rhodes semisimple semigroups

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