The matrix envelope of a semigroup

Abstract

Various representations of semigroups by means of matrices over a group with zero have been thoroughly studied in semigroup theory. It is well known that any semigroup can be represented faithfully by means of the Schiitzenberger representations of some containing semigroup. The representing matrices are either row-monomial, that is, have at most one non-vanishing entry in each row, or column-monomial. While the Rees representation of a semigroup has yielded much useful information, the Schtitzenberger representations of a semigroup S are often so much larger than S itself that little information about S can be obtained from its Schiitzenberger representations. Clearly, one needs to embed S into a matrix semigroup more closely related to S. The purpose of this paper is to faithfully represent a semigroup S which satisfies the descending chain condition on its left and right ideals by rowmonomial A x A matrices over a group with zero. If S has only a finite number of left and right ideals then [A[ is finite, where IAl denotes the cardinal number of the set A. If S has A distinct left and right ideals where A is infinite, then IAl = /Al. From this theorem, it is easy to conclude that a finite semigroup can be embedded in a finite regular semigroup. Throughout this paper, S will be a semigroup which has a zero element and satisfies the descending chain condition on left and right ideals. S’ will denote the semigroup Su 1 arising from S by the adjunction of an identity element 1, unless S already has an identity, in which case S’ = S. If A and B are sets, A B will denote the set of all elements of A which are not in B.