Generating Sets of Completely 0-simple Semigroups

A formula for the rank of an arbitrary finite completely 0-simple semigroup, represented as a Rees matrix semigroup M[G; I,Λ;P ] , is given. The result generalises that of Ruškuc concerning the rank of connected finite completely 0-simple semigroups. The rank is expressed in terms of |I| , |Λ| , the number of connected components k of P and a number rmin which we define. We go on to show that the number rmin is expressible in terms of a family of subgroups of G the members of which are in one to one correspondence with, and determined by the non-zero entries of, the components of P . A number of applications are given, including a generalisation of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B(G, {1, . . . , n}) .