Topic

# Cancellative semigroup

About: Cancellative semigroup is a(n) research topic. Over the lifetime, 1320 publication(s) have been published within this topic receiving 13319 citation(s).

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01 Mar 1984

TL;DR: Gilmer's "Commutative Semigroup Rings" as mentioned in this paper was the first exposition of the basic properties of semigroup rings, focusing on the interplay between semigroups and rings, thereby illuminating both of these important concepts in modern algebra.

Abstract: "Commutative Semigroup Rings" was the first exposition of the basic properties of semigroup rings. Gilmer concentrates on the interplay between semigroups and rings, thereby illuminating both of these important concepts in modern algebra.

427 citations

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TL;DR: In this paper, the authors consider a semigroup satisfying the property abc = ac and prove that it is left semi-normal and right quasi-normal, where ac is the number of variables in the semigroup.

Abstract: This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi- normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular. I. PRELIMINARIES AND BASIC PROPERTIES OF REGULAR SEMIGROUPS In this section we present some basic concepts of semigroups and other definitions needed for the study of this chapter and the subsequent chapters. 1.1 Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S 1.2 Definition: A semigroup (S, .) is rectangular if it satisfies the identity aba = a for all a,b in S. 1.3 Definition: A semigroup (S, .) is called left(right) regular if it satisfies the identity aba = ab (aba = ba) for all a,b in S. 1.4 Definition: A semigroup (S, .) is called regular if it satisfies the identity abca = abaca for all a,b,c in S 1.5 Definition: A semigroup (S, .) is said to be total if every element of Scan be written as the product of two elements of S. i.e, S 2

244 citations

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TL;DR: Some propertes of fuzzy ideals and fuzzy biideals of a semigroup are given and a regular semigroup is characterized that is a semilattice of groups in terms of fuzzy left, right and two-sided ideals and biideal.

Abstract: We give some propertes of fuzzy ideals and fuzzy biideals of a semigroup and characterize a regular semigroup, a semigroup that is a semilattice of left (right) simple semigroups, a semigroup that is a semilattice of left (right) groups and a semigroup that is a semilattice of groups in terms of fuzzy left, right and two-sided ideals and biideals.

188 citations

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02 Dec 2010

TL;DR: In this article, the authors studied the structure of the second dual of the semigroup algebra and its amenability constant, showing that there are 'forbidden values' for this constant.

Abstract: Let $S$ be a (discrete) semigroup, and let $\ell^{\,1}(S)$ be the Banach algebra which is the semigroup algebra of $S$. The authors study the structure of this Banach algebra and of its second dual. The authors determine exactly when $\ell^{\,1}(S)$ is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are 'forbidden values' for this constant. Table of Contents: Introduction; Banach algebras and their second duals; Semigroups; Semigroup algebras; Stone-?ech compactifications; The semigroup $(\beta S, \Box)$; Second duals of semigroup algebras; Related spaces and compactifications; Amenability for semigroups; Amenability of semigroup algebras; Amenability and weak amenability for certain Banach algebras; Topological centres; Open problems; Bibliography; Index of terms; Index of symbols. (MEMO/205/966)

169 citations

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166 citations