Irreducible Semigroups Research Articles

Representations of certain compact semigroups by 𝐻𝐿-semigroups

An HL-semigroup is defined to be a topological semigroup with the property that the Schützenberger group of each H \mathcal {H} -class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HL-semigroups to separate points of S; or equivalently, is S isomorphic to a strict projective limit of HL-semigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HL-semigroups to separate points of S.

An L¹-algebra for algebraically irreducible semigroups

An L¹-algebra for algebraically irreducible semigroups

Totally ordered 𝐷-class decompositions

Introduction. An irreducible semigroup is a compact connected semigroup with identity which contains no proper compact connected subsemigroup containing the identity and meeting the minimal ideal. Since every compact connected semigroup with identity has an irreducible subsemigroup joining its identity to its minimal ideal, the study of the category of irreducible semigroups is basic in any quest for information about the structure of compact semigroups with identity. In a previous announcement [4], the authors described the notion of a hormos and conjectured that an irreducible semigroup is an irreducible hormos. (The structure of the latter was completely described—without proofs—in that announcement.) We also announced a number of categories of semigroups for which we are able to obtain this result. One distinguishing feature of an irreducible hormos is that the Dclass (and in fact even the u-class) decomposition is a totally ordered semigroup—in fact an /-semigroup [6]. I t may be conjectured that a study of semigroups whose P-class decomposition is totally ordered might lead to a description of all irreducible semigroups, and it is this motive which brings us to the results of this announcement. We shall give a brief outline of the techniques we use to prove our results. The details will appear in our forthcoming book. For terminology, the reader is referred to [ l ] and [4],

Algebraically irreducible semigroups

Algebraically irreducible semigroups

Some Generalizations Of Burnsides Theorem

1. Introduction. Burnside's Theorem in the theory of group representations states that a necessary and sufficient condition that a semigroup of matrices of degree n over the complex field be irreducible is that the semigroup contain n2 linearly independent matrices. In the course of dealing with sets of matrices with coefficients in a division ring, Brauer (1) obtained a generalization of this theorem which concerned irreducible semigroups with elements in a division ring.