A new representation theorem for completely [O-] simple semigroups

A semigroup is completely [O-] simple iff it is isomorphic to a semigroup ϕ of binary Boolean (I×I)-matrices (where I is a set) such that for every p,q∈I there exists precisely one matrix (a_ij)∈ϕ such that a_pg=1. Every such isomorphic representation of a completely [O-] simple semigroup is an arbitrary inflation of a reduced representation and all reduced representations are equivalent. Dedicated to Lazar Matveevič Gluskin on the occasion of his fiftieth birthday.