Matrix Representations of ¿-simple Semigroups

By a 'representation' we shall mean throughout a representation by n x n matrices with entries from an arbitrary (commutative) field. Clifford has constructed all representations of completely simple semigroups [1; 4]. Munn has determined the representations of finite semigroups for which the corresponding semigroup algebra is semi-simple [6]. It is noted by Clifford and Preston [4] that if S is a semigroup satisfying the descending chain condition for principal ideals and such that every 0-simple principal factor is completely 0-simple then all the irreducible representations of S can be expressed in terms of those of subgroups of S. This statement is a consequence of work of Clifford [1] and Munn [4]. Munn has determined the irreducible representations of intraregular inverse semigroups [7]. In §1, we prove the following result: Let S be a semigroup having a maximal group homomorphic image G. Then there is a one-to-one correspondence between the representation of G and the nonsingular representations of S which preserves equivalence, reduction and decomposition (Theorem 1.2). An application of this result to inverse semigroups is given. Stoll [11] gives some examples of semigroups with maximal group homomorphic images. In §2, we determine the representations of an important class of ¿¿-simple semigroups by utilizing Theorem 1.2. Let S be a semigroup satisfying the following conditions : (Al) S is d-simple. (A2) S has an identity element. (A3) Any two idempotents of S commute. It is shown by Clifford [2] that the structure of S is determined by that of its right unit semigroup P and that P has the following properties : (Bl) The right cancellation law holds in P. (B2) P has an identity element, 1. (B3) The intersection of two principal left ideals of P is a principal left ideal. Condition (B3) implies that for any a, b in P there exists x and y in P such that