Matrix Representations of Inverse Semigroups

Abstract

In his paper [1], W. D. Munn determines the irreducible matrix representations of an arbitrary inverse semigroup. Munn also gives a necessary and sufficient condition upon a 0-simple inverse semigroup for it to have a non-trivial matrix representation and for such semigroups gives a complete account of their representations. Munn's results rest upon the earlier work of Clifford [2] in which the representations of Brandt semigroups were determined. An alternative account of such representations was given by Munn in [3]. This earlier work is presented in Sections 5.2 and 5.4 of [4].In this paper we obtain a complete determination of the matrix representations of inverse semigroups. We restrict ourselves to inverse semigroups with a zero, and there is clearly no loss in generality in so doing. We show that any representation (without a null component) decomposes into what we term primitive components. Each primitive component in turn decomposes into representations which are determined by representations of certain associated Brandt semigroups. The set of Brandt semigroups involved is determined by what we call the representation ideal series of the representation.Conversely, representation ideal series are abstractly characterized and it is shown that starting with a representation ideal series and the Brandt semigroups it determines then representations of these Brandt semigroups determine, in a unique fashion, a representation of the original semigroup.The methods used are a development of those used by Munn in [1]. The results of Munn may be easily inferred from our results.The terminology and notation followed will be that of [4]. Certain differences from the terminology already used by Munn in earlier work are adopted for systematic reasons in conformity with [10]. Concepts and terminology not in [4] will be defined.The main theorem of this paper was announced in [13].

Exclusive semigroups

“…We shall next investigate the structure of basic primitive exclusive semigroups T. Since the set B of idempotents of T is a primitive regular subsemigroup of T, it follows from Preston [4] that there exists a family {B y :yeF} of rectangular subsemigroups B y of B such that (i)…”

“…However, in general, for any two idempotents e,/of an exclusive semigroup at least one of efandfe is an idempotent. Further, both a commutative semigroup and a completely non-commutative semigroup (4) satisfy the condition (3.1). Moreover, it is easily seen that in an exclusive semigroup S whose idempotents form a band, for any x, ye S the relation (xy) 2 = x 2 y 2 is satisfied except in the case x 2 = e, y 2 = / , ef = e or f, e # / .…”

Primitive homomorphic images of semigroups

“…We repeat firstly some definitions from [2]. A semigroup S = S° is pj" is the congruence on S generated by the relation p x .…”

“…In his paper [2], Preston has shown (lemma 3) that a necessary condition for a regular semigroup to have a O-restricted primitive homomorphic image is that the semigroup be categorical at zero, and (theorem 5) that for an inverse semigroup, this condition is also sufficient. (The terms "primitive", "categorical at zero", and "O-restricted" are defined below.)…”

The structure of pseudo-inverse semigroups

“…II,16,39,44,46] Key words and phrases. Pseudo-inverse semigroup, proper inverse semigroup, rectangular band of inverse semigroups, pseudo-semilattice, semilattice, order automorphism.…”