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Showing papers on "Bicyclic semigroup published in 1997"


Journal ArticleDOI
TL;DR: 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, which is isomorphic to the fundamental groups of the complex KS.
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.

56 citations


Journal ArticleDOI
TL;DR: In this article, the concept of quasi-distribution semigroups on Banach spaces was introduced and a functional calculus for the generator A of a quasi distribution semigroup was introduced.

43 citations


Journal ArticleDOI
TL;DR: In this article, a comparison theorem between two semigroups: a semigroup acting on scalar valued functions and a semi-group acting on vector valued functions is given by the abstract Kato theorem.

36 citations


Book ChapterDOI
01 Sep 1997
TL;DR: The aim of this paper is to present algorithms to compute finite semigroups, given by a set of generators taken in a larger semigroup, called the “universe”.
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.

31 citations


Journal Article

28 citations


Journal ArticleDOI
TL;DR: A more general definition of an HNN extension is introduced and it is shown that free inverse semigroups and the bicyclic semigroup are HNN extensions of semilattices as examples of the authors' new construction.
Abstract: Originally the concept of an HNN extension of a group was introduced by Higman, Neumann and Neumann in their study of embeddability of groups. Howie introduced the concept of HNN extensions of semigroups and showed embeddability in the case that the associated subsemigroups are unitary. On the other hand, T. E. Hall showed the embeddability of HNN extensions of inverse semigroups of a special type in his survey article on amalgamation of inverse semigroups. We introduce a more general definition of an HNN extension and show that free inverse semigroups and the bicyclic semigroup are HNN extensions of semilattices as examples of our new construction. We discuss weak HNN embeddability in several classes of semigroups and strong HNN embeddability in the class of inverse semigroups. One of our main purposes in the study of HNN extensions of inverse semigroups is to employ HNN extensions to examine some algorithmic problems. We prove the undecidability of Markov properties of finitely presented inverse semigroups using HNN extensions. This result was announced by Vazhenin in 1978, but no proof of it has been published to date. We also show undecidability of several non-Markov properties and discuss some undecidable problems on finitely generated inverse subsemigroups of finitely presented inverse semigroups.

25 citations


Journal ArticleDOI

14 citations


Journal ArticleDOI
TL;DR: The main result of as mentioned in this paper is that every finite semigroup is isomorphic to a transitive semigroup of binary relations on a finite set, and that all finite semigroups have faithful transitive representations by 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.

7 citations



Journal ArticleDOI
TL;DR: In this paper, the authors studied some properties of positive quasi-orders on simigroups and used these results to describe all semilattice and chain homomorphic images of 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.

5 citations


01 Jan 1997
TL;DR: In this paper, it was shown that these properties are also true in UG for a large class of locally compact groups, and 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.
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.



Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of finding a non-trivial semigroup, analytic on the right half-plane, which is bounded on the line 1 + it.
Abstract: Introduction The motivation for the work in this paper is a question posed by J. Esterle in [B]. Recall, an analytic semigroup in a Banach algebra B is a B-valued analytic function a defined on the right half-plane H = {z ∈ C : 0} such that a = aa for all z, w ∈ H. The question is the following: If the group algebra (of a non-discrete locally compact group) contains a non-trivial semigroup, analytic on the right half-plane, which is bounded on the line 1 + it, must the group be compact? This question arises from the study of the various semigroups which are to be found in group algebras. More precisely, the type of growth that an analytic semigroup may have on different subsets of H has been discovered to influence the structure of the Banach algebras where the semigroup takes its values, group algebras in particular (see [S] and also [B], [G1], [G2], [G3], [W]). As an example, and related to his classification of radical Banach algebras, Esterle observed that, thanks to the Alfhors-Heins theorem in one complex variable, there cannot exist such algebras containing at the same time semigroups a of polynomial growth on vertical lines in H ([B, pp. 4-65]). This property of the radical algebras had indeed been previously used by Esterle himself in his complex-variable proof of the Wiener tauberian theorem (see [S] for an extended version of it), together with the fact that the group algebra L1(R) has an abundance of such semigroups. For, in this way, L1(R) cannot have any non-trivial radical quotient, whence the Wiener theorem holds. The most important examples of analytic semigroups in L1(R) with polynomial growth on vertical lines are the classical ones. Let us recall that the Gaussian semigroup G(x) = (πz)− n 2 e −|x|2 z , (x ∈ R, 0), is O(|t|n2 ) , whereas the Poisson semigroup P (x) = Γ( 2 )π − 2 z(z + |x|2)−n+1 2 is O(|t| 2 ), on 1 + it as |t| → ∞, for n ≥ 2. From this we might expect that ‖P ‖1 would be O(1), as |t| → ∞, if n = 1 but, instead of this, we get O(log |t|), as is well known (see [S]). Moreover, there was no known example of a non-zero semigroup a in L1(R) bounded on 1 + it, (t ∈ R), and the same can be said for more general groups than R such as stratified Lie groups or non-compact groups with polynomial growth, for instance.


Journal ArticleDOI
TL;DR: In this paper, the problem of representing a finite inverse semigroup by partial transformations of a graph is treated, and the notions of weighted graph and its weighted partial isomorphisms are introduced.
Abstract: In the paper, the problem of representing a finite inverse semigroup by partial transformations of a graph is treated. The notions of weighted graph and its weighted partial isomorphisms are introduced. The main result is that any finite inverse semigroup is isomorphic to the semigroup of weighted partial isomorphisms of a weighted graph. This assertion is a natural generalization of the Frucht theorem for groups.

01 Jan 1997
TL;DR: In this article, a multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties is described both semantically and syntactically.
Abstract: A multiplication of e-varieties of regular $E$-solid semigroups by inverse semigroup varieties is described both semantically and syntactically. The associativity of the multiplication is also proved.

Journal ArticleDOI
TL;DR: In this article, it was shown that the efficiency of the symmetric inverse monoid on a set of n elements can depend solely on the index of nilpotence of the rankn − 1 generator, and various ways of expressing elements of the semigroup most efficiently as a product of generators.
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.


Journal Article
TL;DR: Semigroups, ideal extensions of a right (left)-zero semigroup by completely simple semigroup with zero adjoint, whose congruences are pairwise permutable are completely determined as discussed by the authors.
Abstract: Semigroups, ideal extensions of a right (left)-zero semigroup by completely simple semigroup with zero adjoint, whose congruences are pairwise permutable are completely determined.

01 Jan 1997
TL;DR: In this article, a generalization of the Reilly's results for generalized inverse ∗-semigroups is presented, in which the semigroup is represented by isomorphisms between principal ideals of the semilattice E(S).
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.

Journal ArticleDOI
TL;DR: In this article, the authors construct a monoid, the group of units of which a semidirect product is a semigroup of endomorphisms, and determine whether such a group is regular, orthodox, or inverse.
Abstract: Given any family of normal subgroups of a group, we construct in a natural way a certain monoid, the group of units of which is a semidirect product. We apply this to obtain a description of both the semigroup of endomorphisms and the group of automorphisms of an Ockham algebra of finite boolean type. We also determine when such a monoid is regular, orthodox, or inverse