Showing papers on "Cancellative semigroup published in 1990"
18 citations
12 citations
12 citations
01 Jan 1990
TL;DR: In this paper, it was shown that a regular semigroup S C 9 (F) has the permutation property 37, m > 2, if for every a,..., am E S, there exists a non-trivial permutation a such that al a = a,()... aU(,nf).
Abstract: It is well-known that if a semigroup algebra K[S] over a field K satisfies a polynomial identity then the semigroup S has the permutation property. The converse is not true in general even when S is a group. In this paper we consider linear semigroups S C 9 (F) having the permutation property. We show then that K[S] has a polynomial identity of degree bounded by a fixed function of n and the number of irreducible components of the Zariski closure of S. A semigroup S is said to have the property 37, m > 2, if for every a,, .. ., am E S, there exists a non-trivial permutation a such that al a = a,()... aU(,nf). S has the permutation property 37 if S satisfies 3Y for some m> 2. The class of groups of this type was shown in [3] to consist exactly of the finite-by-abelian-by-finite groups. For the recent results and references on this extensively studied class of groups, we refer to [1]. The above description of groups satisfying 37 was extended to cancellative semigroups in [11], while a study of regular semigroups with this property was begun in [6]. In connection with the corresponding semigroup algebras K[S] over a field K, the problem of the relation between the property 37 for S and the Plproperty for K[S] attracted the attention of several authors. It is straightforward that S has 37 whenever K[S] satisfies a polynomial identity. However the converse fails even for groups in view of [3] and the characterization of PI group algebras, cf. [1 5]. On the other hand, K[S] was shown to be a PI-algebra whenever S is a finitely generated semigroup (satisfying 3 ) of one of the following types: periodic [20], cancellative [11], 0-simple [3, 5], inverse, or a Rees factor semigroup of free semigroup, cf. [12]. However, a finitely generated regular semigroup S with two non-zero OF-classes having Y but with K[S] not being PI was constructed in [12]. The main result of this paper is that if S is a linear semigroup satisfying 39, then K[S] is PI for any field K. In the course of the proof, we obtain a structural description of a strongly 7r-regular semigroup of this type. The basic technique is to consider the Zariski closure S of S. Then S is a linear Received by the editors July 7, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20M25, 16A38; Secondary 20M20, 16A45. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page
6 citations
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3 citations
01 Jan 1990
TL;DR: A survey of results on the Jacobson radical of the semigroup ring of a completely regular semigroup over a ring with unity is given in this article, where a semigroup S is said to be completely regular if and only if it is covered by its subgroups.
Abstract: A semigroup S is said to be completely regular if and only if it is covered by its subgroups; that is, if and only if, for each a ∈ S, a ∈ a2 S∩S a2. Groups and bands (semigroups of idempotents) are extreme special cases. In this paper a survey is given of results on the Jacobson radical of the semigroup ring of a completely regular semigroup over a ring with unity. Much of the inspiration is derived from the study of group rings, in which a similar interplay of two distinct branches of algebra is apparent. The work discussed covers a period of some thirty- six years, from the first paper on semigroup rings by Marianne Teissier (1952) to the present day.
3 citations
TL;DR: In this article, the authors define the semigroup of an ordinary multiple point of an analytic plane curve f. This semigroup on tuples of integers is completely characterised in terms of the order of f, that is the number of distinct tangent directions to f at that point.
Abstract: In this paper we define the semigroup of an ordinary multiple point of an analytic plane curve f. This semigroup on tuples of integers is completely characterised in terms of the order of f, that is the number of distinct tangent directions to f at that point.
3 citations
TL;DR: A ring (R,*) with involution * is called formally complex if implies that all Ai are 0 as discussed by the authors, and a semigroup ring (S, *) with proper involution is a formally complex ring.
Abstract: A ring (R, *) with involution * is called formally complex if implies that all Ai are 0. Let (R, *) be a formally complex ring and let S be an inverse semigroup. Let (R[S], *) be the semigroup ring with involution * defined by . We show that (R[S], *) is a formally complex ring. Let (S, *) be a semigroup with proper involution *(aa* = ab* = bb* ⇒ a = b) and let (R, *′) be a formally complex ring. We give a sufficient condition for (R[S], *′) to be a formally complex ring and this condition is weaker than * being the inverse involution on S. We illustrate this by an example.
2 citations
TL;DR: In this paper, a simple proof is given that the infinitesimal generator of the heat semigroup and the Poisson semigroup are scalar operators in Lp (Rn), 1
TL;DR: In this article, the authors studied combinatorial properties of an infinite word on a two-letter alphabet in order to provide a solution to a problem of Semigroup Theory, which they called Semigroup Problem.
Abstract: We study some combinatorial properties of an infinite word on a two letters alphabet in order to provide a solution to a problem of Semigroup Theory.