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Cancellative semigroup
About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.
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TL;DR: In this paper, it was shown that the semigroup algebra of a commutative ring with unit inverse semigroups can be described as a convolution algebra of functions on the universal \'etale groupoid associated to the inverse semigroup.
Abstract: Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality.
Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup $S$ that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.
4 citations
TL;DR: In this article , it was shown that if char(D ) = 0 (resp., char( D ) = p > 0), then D [Γ] is a weakly Krull domain if and only if D is an integral domain and G is a torsion-free commutative cancellative semigroup with identity element and quotient group G .
Abstract: . Let D be an integral domain and Γ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group G . In this paper, we show that if char( D ) = 0 (resp., char( D ) = p > 0), then D [Γ] is a weakly Krull domain if and only if D is a weakly Krull UMT-domain, Γ is a weakly Krull UMT-monoid, and G is of type (0 , 0 , 0 ,... ) (resp., type (0 , 0 , 0 ,... ) except p ). Moreover, we give arithmetical applications of this result. Our results show that there is also a class of weakly Krull domains, which are not Krull but have full system of sets of lengths.
4 citations
4 citations
4 citations
TL;DR: For an arbitrary set X (finite or infinite), the symmetric inverse semigroup of partial injective transformations on X is defined in this article, and the structure of C(a) is determined in terms of Green's relations.
Abstract: For an arbitrary set X (finite or infinite), denote by I(X) the symmetric inverse semigroup of partial injective transformations on X. For an element a in I(X), let C(a) be the centralizer of a in I(X). For an arbitrary a in I(X), we characterize the elements b in I(X) that belong to C(a), describe the regular elements of C(a), and establish when C(a) is an inverse semigroup and when it is a completely regular semigroup. In the case when the domain of a is X, we determine the structure of C(a) in terms of Green's relations.
10.1017/S0004972712000779
4 citations