Showing papers on "Cancellative semigroup published in 1997"

On the algebra of semigroup diagrams

Abstract: In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid of reduced semigroup diagrams over the presentation S = may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

Algorithms for computing finite semigroups

Abstract: The aim of this paper is to present algorithms to compute finite semigroups. The semigroup is given by a set of generators taken in a larger semigroup, called the “universe”. This universe can be for instance the semigroup of all functions, partial functions, or relations on the set , or the semigroup of × n matrices with entries in a given finite semiring.

Semigroups which contain magnifying elements are factorizable

Abstract: A semigroup S is factorizable if it contains two proper subsemigroups A and B such that S = AB. An element a of a semigroup 5 is a left ( resp. right) magnifier if there exists a proper subset M of S such that S = aM (resp. S - Ma). In this paper we prove that every semigroup containing magnifying elements is factorizable. Thus we solve a problem raised up by F. Catino and F. Migliorini in [2], namely to find necessary and sufficient conditions in order that a semigroup with magnifying elements be factorizable. Partial answers to this problem have been obtained by K. Tolo ([14]), F. Catino and F. Migliorini ([2]), for semigroups with left magnifiers and which are regular or have left units or right magnifiers, by V. M. Klimov ([9]), for Baer-Levi and Croisot-Teissier semigroups, and by M. Gutan ([4]), for right cancellative, right simple, idempotent free semigroups.

Every semigroup is isomorphic to a transitive semigroup of binary relations

Abstract: Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set). Let BA be the set of all binary relations between elements of a set A. We consider BA as a semigroup with the operation of relative product o. Its subsemigroups are called semigroups of binary relations. A (faithful) representation of a semigroup S by relations is a(n injective) homomorphism of S into BA, A being any set. A subset 4' C BA is called transitive if U 4I = A x A (that is, for any a, b C A there exists o c 4' with (a, b) c ep). A representation P of a semigroup S is called transitive if P(S) is a transitive set of relations, that is, P can be viewed as a homomorphism of S onto a transitive semigroup of relations. A longstanding problem of semigroup theory (see [4]) asks which semigroups have faithful transitive representations by relations. An equally longstanding conjecture is: all. Various classes of semigroups (subdirectly irreducible, with zero, completely [0]-simple) were proved to have faithful transitive representations by relations (see [4] and [5]). The main results of this paper are the following theorems. Theorem A. Every semigroup is isomorphic to a transitive semigroup of binary relations. Theorem B. Every finite semigroup is isomorphic to a transitive semigroup of binary relations on a finite set. Before proving the theorems, we mention some open problems. Open Problems. 1. A relation o C A x A is called a multipermutation if its domain and range coincide with A, that is, if, given any a c A, there exist b, c E A with (a, b), (c, a) c ep. Every semigroup is isomorphic to a semigroup of multipermutations [4]. Which semigroups are isomorphic to transitive semigroups of multipermutations? 2. Every set 4I of binary relations is ordered by the inclusion relation c, and every semigroup (D; o) of relations becomes an ordered semigroup (1; o; C). Speaking of orders, we always mean partial orders. Clearly, c is a stable order on 4I (that is, c is a subsemigroup of the semigroup 4I x 4I or, equivalently, Received by the editors September 20, 1995. 1991 Mathematics Subject Classification. Primary 20M30, 20M10; Secondary 03G15, 04A05, 05C12, 08A02, 20M12, 20M20.

On seminormal semigroups

Abstract: Each semigroup, written additively, is a non-zero, commutative, torsion-free cancellative semigroup with 0. Let S be a semigroup with the quotient group G. T is called an oversemigroup of S if T is a subsemigroup of G containing S. An element \( t\in T \) is said to be integral over S if \( nt \in S \) for some positive integer n. The set S' of elements \( t \in G \) that are integral over S is called the integral closure of S. We say that S is a seminormal semigroup if \( 2\alpha, 3\alpha \in S \) for \( \alpha \in G \), we have \( \alpha \in S \). Also, S is called a valuation semigroup if either \( \alpha \in S \) or \( -\alpha \in S \) for each \( \alpha \in G \). An ideal of S is a non-empty subset I of S such that \( I \supset s + I \) for each \( s\in S \). An ideal I of S is prime if \( I e S \) and if \( x + y \in I \) implies \( x \in I \) or \( y \in I \) for \( x,y \in S \). We say that dim (S) = 1 if S has one and only one prime ideal. In this paper, we shall consider the properties of seminormal semigroups. In particular, we shall prove that, for a semigroup S with dim S = 1, (1) If S is seminormal, then each integral oversemigroup of S is seminormal. (2) Each oversemigroup of S is seminormal if and only if both S is seminormal and S' is a valuation semigroup.

The lattice of positive quasi-orders on a semigroup

Abstract: In the present paper we study some properties of positive quasi-orders on simigroups and using these results we describe all semilattice and chain homomorphic images of a semigroup.

On some semigroup compactifications

Abstract: The LUG-compactification UG of a locally com­ pact group is a semigroup with an operation which extends that of G and which is continuous (only) in one variable. When G is discrete, UG and the Stone-Cech compactification {3G are identi­ cal. Some algebraic properties, such as the num­ ber of left ideals and cancellation, are known to hold in the semigroup {3N where N is the additive semigroup of the integers. We show that these properties are also true in UG for a large class of locally compact groups. The method used is to transfer the information from {3N to (3G where G is an infinite discrete group (or a cancellative commutative semigroup), and then to UG where G is not necessarily discrete.

The universal nilpotent group compactification of a semigroup

Abstract: The purpose of this paper is to introduce an algebra of functions on a semitopological semigroup and to study these functions from the point of view of universal semigroup compactification. We show that the corresponding semigroup compactification of this algebra is universal with respect to the property of being a nilpotent group. The general approach to the theory of semigroup compactification is based on the Gelfand-Naimark theory of C∗-algebras of functions. There are many papers which deal with the characterization of certain universal semigroup compactifications in terms of function algebras. Seminal work in this context was done by K. de Leeuw and I. Glicksberg [3]. The universal group compactification is given by Junghenn [6], in terms of some types of distal functions. This paper deals with the construction of a function algebra on a semitopological semigroup, used to characterize the universal nilpotent group compactification of it, and investigating some of its properties. First we recall some preliminaries. Throughout this paper, S shall be a semitopological semigroup, unless otherwise mentioned. For notation and terminology we shall follow Berglund et al. [2] as far as possible. Thus a semigroup compactification of S is a pair (ψ,X), where X is a compact, Hausdorff, right topological semigroup and ψ : S → X is a continuous homomorphism with dense image such that for each s ∈ S, the mapping x 7→ ψ(s)x : X → X is continuous. The C∗-algebra of all continuous bounded complex-valued functions on a topological space Y is denoted by C(Y ). For C(S) left and right translations Ls and Rt are defined for all s, t ∈ S and f ∈ C(S) by (Lsf)(t) = f(st) = (Rtf)(s). A left translation invariant C∗-subalgebra F of C(S) (i.e. Lsf ∈ F for all s ∈ S and f ∈ F ) containing the constant functions is called m-admissible if the function s 7→ (Tμf)(s) = μ(Lsf) is in F for all f ∈ F and μ ∈ S (=the spectrum of F ); then the product of μ, ν ∈ S can be defined by μν = μ ◦ Tν and the Gelfand topology on S makes ( , S ) a semigroup compactification (called the F -compactification) of S, where : S → S is the evaluation mapping. The reader is referred to sections 3.1 and 3.3 of [2] for the one-to-one correspondence between compactifications of S and m-admissible subalgebras of C(S), and also for a discussion of universal P -compactifications, whose existence for a wide variety of properties P is given in terms of subdirect products. Received by the editors January 16, 1996. 1991 Mathematics Subject Classification. Primary 22A20, 43A60.

Umbral Calculus and Cancellative Semigroup Algebras

Abstract: We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures). Systematic usage of cancellative semigroup, their convolution algebras, and tokens between them provides a common language for description of objects from these three fields. Keywords: cancellative semigroups, umbral calculus, harmonic analysis, token, convolution algebra, integral transform

Umbral Calculus: a Model from Convoloids

Abstract: We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures). Systematic usage of cancellative semigroup, their convolution algebras, and tokens between them provides a common language for description of objects from these three fields. Keywords: cancellative semigroups, umbral calculus, harmonic analysis, token, convolution algebra, integral transform

Semigroups presented by one relation and satisfying the Church-Rosser property

Abstract: This paper is devoted to the study of semigroups presented by a single defining relationA=B and satisfying the Church-Rosser property.

Hierarchy of efficient generators of the symmetric inverse monoid

Abstract: It is well-known that the symmetric inverse monoid on a set ofn elements can be generated as a semigroup by its group of units and a single element of rankn − 1. We show that the efficiency with which the semigroup is generated in this way depends solely on the index of nilpotence of the rankn − 1 generator. We also investigate the various ways of expressing elements of the semigroup most efficiently as a product of generators.

Some Remarks on Representations of Generalized Inverse [*-]Semigroups

Abstract: The Munn representation of an inverse semigroup S, in which the semigroup is represented by isomorphisms between principal ideals of the semilattice E(S), is not always faithful. By introducing a concept of a presemilattice, Reilly considered of enlarging the carrier set E(S) of the Munn representation in order to obtain a faithful representation of S as an inverse subsemigroup of a structure resembling the Munn semigroup TE(S). The purpose of this paper is to obtain a generalization of the Reilly’s results for generalized inverse ∗-semigroups.

Symmetric identities in graded algebras

Abstract: Let P k be the symmetric polynomial of degree k ie, the full linearization of the polynomial x k Let G be a cancellation semigroup with 1 and R a G-graded ring with finite support of order n We prove that if R 1 satisfies $ P_k \equiv 0 $ then R satisfies $ P_{kn} \equiv 0 $