Affine semigroups

Cohen, Haskell; Collins, H. S.

doi:10.1090/s0002-9947-1959-0107677-x

Cited by 17 publications

(22 citation statements)

References 9 publications

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“…From this, one sees that the class of pseudoinvertible semigroups includes all semigroups admitting relative inverses, all periodic semigroups; all semigroups of matrices; all finite dimensional affine semigroups (for definition of an afRne semigroup, see [2]) and many others. Proof.…”

Section: Theorem L β If S Is Discrete Then (S T)~ Is Compactmentioning

confidence: 99%

Generalized character semigroups: The Schwarz decomposition

Lin¹

1965

Pacific J. Math.

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The author's r€sum §: A structure theorem due to S. Schwarz asserts that if S is a finite abelian or a compact abelian semigroup admitting relative inverses, than the character semigroup of S is decomposed into a disjoint union of character groups of certain maximal subgroups of S. In this note, among other things, we generalize this Schwarz Decomposition Theorem to a broader class of semigroups, the so-called pseudo-invertible semigroups. We also relax the range of the characters from the semigroup of complex numbers to a more general semigroup.For notations and terms not defined here see A. D β Wallace [11]. Throughout this paper, let S be always a compact commutative semigroup, unless otherwise stated. By a character of S is meant a continuous homomorphism of S into the multiplicative semigroup C of the complex numbers endowed with the usual Euclidean topology. The collection of all characters of S, with the value-wise multiplication of functions, endowed with the compact-open topology, forms a semigroup which will be denoted by (S, C)~ or simply S~, and will be called the character semigroup of S. Hewitt and Zuckerman [4] use the term semicharacter, in the discrete case, for not identically zero characters. Here we use (S, C) or simply S, as distinguished from SΓ, to denote the collection of semicharacters of S. We note that S, in general, need not be a semigroup. We first draw attention to the fact that if χ is a character of S, then | χ(x) | ^ 1 for every x in S o For, otherwise χ(S) would not be compact. Thus, in the study of characters, only the unit disc {z: | z | ^ 1} of the complex numbers is used. Let us write D for this unit disc. The set D itself forms an important semigroup which is compact, connected, commutative, cancellable, 1 has zero 0 and unit 1; moreover the circumference {z: \ z | = 1} of D is the maximal subgroup H(l) and D\H (1) is an ideal. However, only some of these are needed as we shall see below.Throughout the rest of this paper, let T be an arbitrary, but fixed, compact commutative cancellable semigroup with zero z and unit u 2 It is to be understood that z ^ u.

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Section: Theorem L β If S Is Discrete Then (S T)~ Is Compactmentioning

confidence: 99%

Generalized character semigroups: The Schwarz decomposition

Lin¹

1965

Pacific J. Math.

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“…(2) There is an associative multiplication defined in S which is jointly continuous in the topology on S inherited from X.…”

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“…By a theorem due to Wendel [2], if S is a compact affine semigroup with identity u, then each point of S with inverse is an extreme point of S. If, conversely, each extreme point has an inverse then the set of extreme points of S is the maximal group of the idempotent u and is, therefore, compact [9]. In this case, we shall say S is group-extremal.…”

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confidence: 99%

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On representations of certain semigroups

Friedberg¹

1966

Pacific J. Math.

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“…A subsemigroup S of the multiplicative semigroup of JS?F will be said to be an affine semigroup over $ if S is a linear variety, i.e., a translate of a linear subspace of ££V.This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.…”

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confidence: 99%

Affine semigroups over an arbitrary field

Clark

1965

Proceedings of the Glasgow Mathematical Association

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Let S£V denote the algebra of all linear transformations on an w-dimensional vector space V over a field . A subsemigroup S of the multiplicative semigroup of JS?F will be said to be an affine semigroup over $ if S is a linear variety, i.e., a translate of a linear subspace of ££V.This concept in a somewhat different form was introduced and studied by Haskell Cohen and H. S. Collins [1]. In an appendix we give their definition and outline a method of describing possibly infinite dimensional affine semigroups in terms of algebras and supplemented algebras.Except in the appendix, V and hence also £PV will be supposed finite-dimensional. In §1 we show that some power of every element of an affine semigroup S lies in a subgroup of S and that S always contains a completely simple minimal ideal K. (For definitions see below.) We then obtain a decomposition of S into a group, an algebra, and two vector spaces.The minimal ideal K of an afflne semigroup is not in general a linear variety, as was observed by Cohen and Collins [1]. When K is not a linear variety, one turns naturally to M(K), the smallest linear variety containing K. We show in §2 that M(K) n -K for any integer n exceeding the dimension of M{K).UK^M(K) and every element ofKis idempotent, then we are able to find a subsemigroup of M{K) isomorphic to the example of Cohen and Collins (see 2.1). In this case we also show that M{K) 2 = K. The requirement that every element of K be idempotent is of some interest since this is always true for affine semigroups on locally convex linear topological spaces which are generated by compact convex subsemigroups (see Theorem 3 of [1] and 2.10 below).The author wishes to express his gratitude to Professor A. H. Clifford for his encouragement and for many useful suggestions during the preparation of this paper.

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Affine semigroups

Cited by 17 publications

References 9 publications

Generalized character semigroups: The Schwarz decomposition

Generalized character semigroups: The Schwarz decomposition

On representations of certain semigroups

Affine semigroups over an arbitrary field

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