GENERATORS AND RELATIONS OF DIRECT PRODUCTS OF SEMIGROUPS

The purpose of this paper is to give necessary and sufficient conditio ns for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequ ence we construct a semigroup S of order 11 such that S x T is finitel y generated but not finitely presented for every finitely generated in finite semigroup TI By way of contrast we show that, if S and T belong to a wide class of semigroups, then S x T is finitely presented if an d only if both S and T are finitely presented, exactly as in the case of groups and monoids.