Irreducible Matrix Representations of Finite Semigroups

Munn [9] has shown that for a semigroup S satisfying the minimal condition on principal ideals, there is a natural one-to-one correspondence between irreducible representations of S and irreducible representations vanishing at zero of its 0-simple (or simple) principal factors; for the case of S finite, see Ponizovskii [11]. On the other hand, Clifford, [3] and [4], has obtained all representations of a completely 0-simple semigroup as "extensions" of those of its maximal subgroups. Combining their results, one can, in principle, obtain all irreducible representations of a semigroup satisfying the minimal conditions on principal left and right ideals and thus of finite semigroups. However, in constructing the representations of a completely 0-simple semigroup S=J(°(G;I, A;F), one has to solve the problem in matrix theory of factoring the block matrix