Cancellative semigroup

About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.

Group ideals in a semigroup of measures

Abstract: In this paper we introduce the notion of a group ideal in a semigroup. We shall prove that all group ideals of a compact affine semigroup are convex and dense. This generalizes many results in the literature concerning ideals in semigroups.

Prime and semiprime semigroup rings of cancellative semigroups

Abstract: Let S be a cancellative semigroup. This paper is motivated by the problem of finding a description of semigroup rings K [ S ] over a field K that are semiprime or prime. Results of this type are well-known in the case of a group ring K [ G ], cf. [8]. The description, as well as the proofs, involve the FC-centre of G defined as the subset of all elements with finitely many conjugates in G . In [4], [5] Krempa extended the FC-centre techniques to the case of an arbitrary cancellative semigroup S . He defined a subsemigroup Δ( S ) of S which coincides with the FC-centre in the case of groups, and can be used to describe the centre and to study special elements of K [ S ]. His results were strengthened by the author in [7], where Δ( S ) was also applied in the context of prime and semiprime algebras K [ S ]. However, Δ( S ) itself is not sufficient to characterize semigroup rings of this type. We note that in [2], [3] Dauns developed a similar idea for a study of the centre of semigroup rings and certain of their generalizations.

Rank properties in finite inverse semigroups

Abstract: Abstract Two possible concepts of rank in inverse semigroup theory, the intermediate I-rank and the upper I-rank, are investigated for the finite aperiodic Brandt semigroup. The so-called large I-rank is found for an arbitrary finite Brandt semigroup, and the result is used to obtain the large rank of the inverse semigroup of all proper subpermutations of a finite set.

Cross-connections of linear transformation semigroup

Abstract: Cross-connection theory developed by Nambooripad is the construction of a semigroup from its principal left (right) ideals using categories. We briefly describe the general cross-connection theory for regular semigroups and use it to study the {normal categories} arising from the semigroup $Sing(V)$ of singular linear transformations on an arbitrary vectorspace $V$ over a field $K$. There is an inbuilt notion of duality in the cross-connection theory, and we observe that it coincides with the conventional algebraic duality of vector spaces. We describe various cross-connections between these categories and show that although there are many cross-connections, upto isomorphism, we have only one semigroup arising from these categories. But if we restrict the categories suitably, we can construct some interesting subsemigroups of the {variant} of the linear transformation semigroup.

Quasi-isometry and finite presentations of left cancellative monoids

Abstract: We show that being finitely presentable and being finitely presentable with solvable word problem are quasi-isometry invariants of finitely generated left cancellative monoids. Our main tool is an elementary, but useful, geometric characterization of finite presentability for left cancellative monoids. We also give examples to show that this characterization does not extend to monoids in general, and indeed that properties such as solvable word problem are not isometry invariants for general monoids.