Maximal subsemigroups of finite semigroups

AbstractIf M is a maximal (proper) subsemigroup of a finite semigroup S, then M contains all but one J-class J(M) of S. When J(M) is non-regular J(M)⊃M= so M=S−J(M). When J(M) is regular either J(M)⊃M= or M⊃J(M) has a natural form with respect to the Green-Rees coordinates in J(M). Specifically, there exist an isomorphism j : J(M)0 → M0(A, B, G, C) of J(M)0 with a Rees regular matrix semigroup so that j(M⊃J(M))=G′×A×B, where G′ is a maximal subgroup of G or j(M⊃J(M)) is the complement of a “rectangle” of H-classes of M0(A, B, G, C). In the first case, (M⊃J(M))0 is a maximal subsemigroup of J(M)0. In the second, (M⊃J(M))0 is maximal in J(M)0 when j(M⊃J(M)) = M0(A, B, G, C) - (G × A′ × B′) for proper subsets A′ and B′ of A and B, respectively, but need not be when j(M⊃J(M))=G×A×B′ or j(M⊃J(M))=G×A′×B