Rings related to completely 0-simple semigroups

Abstract

Communicated by G. B. PrestonA.H. Clifford ([2], [3]) has shown that all finite dimensional irreducible representations of a completely 0-simple semigroup can be obtained as extensions of those of its maximal subgroups. Lallement and Petrich,[7], have given an alternative method for constructing the irreducible representations of a finite 0-simple semigroup from its Schutzenberger representation ([13]). Using the form which they obtain for the irreducible representations of a finite 0-simple semigroup S = Jt*(G; m, n; P), Lallement and Petrich show thatwhere ~# is the Munn algebra isomorphic to the (contracted) algebra $(S) of S over $.In the same paper, necessary and sufficient conditions are given for the existence of an isomorphism when 0 is algebraically closed and t is the invertibility rank of P. However the general problem of determining the structure of <P(S)/Rad <P(S) is not solved there nor is the case of infinite completely 0-simple semigroups considered. The main object of this paper is to determine the radical of <P(S) when S is an arbitrary completely 0-simple semigroup and also to obtain a representation for #(S)/Rad #(S) as a subdirect sum of computable primitive rings.The representation theory developed in [2], [3] depends strongly on the fact that the representations under consideration are finite dimensional and it is not adequate to deal with the infinite dimensional case. In [9] we have described a variant of Clifford's theory which determines all representations of S « ^°(G; I, A; P), by endomorphisms of 9?-modules, in terms of proper representations of G. This representation theory was developed for a class of rings which we call Munn rings. These rings are the infinite dimensional analogs of the Munn algebras considered in [11], Section 4; thus the algebra of any completely 0-simple semigroup is a Munn ring.The results of [9] provide the tools with which to investigate the ideal struc-257

Semigroups of rectangular matrices under a sandwich operation

“…(Here, the sandwich matrix A arises from the celebrated Rees structure theorem [81] for completely 0-simple semigroups.) Since their introduction in [68], Munn rings have been studied by numerous authors, and continue to heavily inflence the theory of semigroup representations: for classical studies, see [12][13][14]37,54,[62][63][64][68][69][70]76]; for modern accounts, see for example [1,29,47,75,78,79,85,86], and especially the monographs [73,74,77,82,87].…”

Complex representations of matrix semigroups

“…Now, for every field K, the contracted semigroup ring KJS] may be identified with the so-called Munn ring Proof. The former assertion is established in [13]. The latter then follows from the semisimplicity of group rings of linear groups in characteristic zero, cf.…”

The Jacobson radical of a band ring

“…The semigroup ring R [S] of S over R is defined as follows: the additive subgroup of R [S] is that of the free unital left /2-module with S as a basis; and multiplication is defined (in terms of the given multiplication in R and S) by the rule Recall that a band S is rectangular if and only if <S = / x J for some non-empty sets / and J, with multiplication (i,j) (»',/) = (»,/) (i,i'el;j,j'eJ). The first part of the following lemma can be deduced from theorem 16 of [4]. For the special case in which R is a field and S is finite, see also [6].…”