Showing papers on "Bicyclic semigroup published in 2009"

The q-theory of Finite Semigroups

Abstract: Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

From right PP monoids to restriction semigroups: a survey

Abstract: Left restriction semigroups are a class of semigroups which generalise inverse semigroups and which emerge very naturally from the study of partial transformations of a set. Consequently, they have arisen in a variety of different contexts, under a range of names. One of the various guises under which left restriction semigroups have appeared is that of weakly left E-ample semigroups, as studied by Fountain, Gomes, Gould and Lawson, amongst others. In the present article, we will survey the historical development of the study of left restriction semigroups, from the `weakly left E-ample' perspective, and sketch out the basic aspects of their theory.

Tight representations of semilattices and inverse semigroups

Abstract: By a Boolean inverse semigroup we mean an inverse semigroup whose semilattice of idempotents is a Boolean algebra. We study representations of a given inverse semigroup \(\mathcal{S}\) in a Boolean inverse semigroup which are tight in a certain well defined technical sense. These representations are supposed to preserve as much as possible any trace of Booleanness present in the semilattice of idempotents of \(\mathcal{S}\) . After observing that the Vagner–Preston representation is not tight, we exhibit a canonical tight representation for any inverse semigroup with zero, called the regular tight representation. We then tackle the question as to whether this representation is faithful, but it turns out that the answer is often negative. The lack of faithfulness is however completely understood as long as we restrict to continuous inverse semigroups, a class generalizing the E*-unitaries.

The rank of the endomorphism monoid of a uniform partition

Abstract: The rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open question.

Biflatness of semigroup algebras

Abstract: We shall study the biflatness of the convolution algebra l 1(S) for a semigroup S. We show that for any semigroup S such that l 1(S) is biflat the canonical partial ordering on the idempotents must be uniformly locally finite. We use this to characterize the biflatness of l 1(S) for an inverse semigroup S.

Towards a better understanding of the semigroup tree

Abstract: In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier by the first author. These regularities admit two different types of behavior and in this work we investigate which of the two types takes place for some well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of the number of semigroups with given genus.

The Bergman property for semigroups

Abstract: In this article, we study the Bergman property for semigroups and the associated notions of cofinality and strong cofinality. A large part of the paper is devoted to determining when the Bergman property, and the values of the cofinality and strong cofinality, can be passed from semigroups to subsemigroups and vice versa. Numerous examples, including many important semigroups from the literature, are given throughout the paper. For example, it is shown that the semigroup of all mappings on an infinite set has the Bergman property but that its finitary power semigroup does not; the symmetric inverse semigroup on an infinite set and its finitary power semigroup have the Bergman property; the Baer�Levi semigroup does not have the Bergman property.

Complexity of the identity checking problem for finite semigroups

Abstract: We prove that the identity checking problem in a finite semigroup S is co-NP-complete whenever S has a nonsolvable subgroup or S is the semigroup of all transformations on a 3-element set. Bibliography: 31 titles.

Cayley automaton semigroups

Abstract: In this paper we characterize when a Cayley automaton semigroup is finite, is free, is a left zero semigroup, is a right zero semigroup, is a group, or is trivial. We also introduce dual Cayley automaton semigroups and discuss when they are finite.

Free adequate semigroups

Abstract: We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial "folding" operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are J-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.

Wandering vectors and the reflexivity of free semigroup algebras

Abstract: A free semigroup algebra S is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering for S if the set of images of x under non-commuting words in the generators of S is orthonormal. We establish the following dichotomy: either a free semigroup algebra has a wandering vector, or it is a von Neumann algebra. Consequences include that every free semigroup algebra is reflexive, and that certain free semigroup algebras are hyper-reflexive with a very small hyper-reflexivity constant.

General theorems on automorphisms of semigroups and their applications

Abstract: We introduce the notion of a strong representation of a semigroup in the monoid of endomorphisms of any mathematical structure, and use this concept to provide a theoretical description of the automorphism group of any semigroup. As an application of our general theorems, we extend to semigroups a well-known result concerning automorphisms of groups, and we determine the automorphism groups of certain transformation semigroups and of the fundamental inverse semigroups.

Inverses of generators of nonanalytic semigroups

Abstract: Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup {e}t≥0. It is shown that A−1 generates an O(1 + τ) A(1 − A)−1-regularized semigroup. Several equivalences for A−1 generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of {e}t≥0, on subspaces, for A−1 generating a strongly continuous semigroup, and to show that the inverse of −d/dx on the closure of its image in L([0,∞)) does not generate a strongly continuous semigroup. We also show that, for k a natural number, if {e}t≥0 is exponentially stable, then ‖e −1 x‖ = O(τ1/4−k/2) for x ∈ D(A).

Criteria for analyticity of subordinate semigroups

Abstract: Let ψ be a Bernstein function. A. Carasso and T. Kato obtained necessary and sufficient conditions for ψ to have the property that ψ(A) generates a quasibounded holomorphic semigroup for every generator A of a bounded C0-semigroup in a Banach space, in terms of some convolution semigroup of measures associated with ψ. We give an alternative to Carasso-Kato’s criterion, and derive several sufficient conditions for ψ to have the above-mentioned property.

On U-orthodox semigroups

Abstract: As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in Open image in new window on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup WU and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups.

On the cayley semigroup of a finite aperiodic semigroup

Abstract: Let S be a finite semigroup. In this paper, we introduce the functions φs:S* → S*, first defined by Rhodes, given by φs([a1,a2,…,an]) = [sa1,sa1a2,…,sa1a2 ⋯ an]. We show that if S is a finite aperiodic semigroup, then the semigroup generated by the functions {φs}s ∈ S is finite and aperiodic.

Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

Abstract: Inspired by the results of Adin, Postnikov and Roichman, we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup and the factor power of the symmetric group. Furthermore, we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the q-rook monoid algebra.

On prime, weakly prime ideals in po-Γ-semigroups

Abstract: It is well known that the ideals of an ordered semigroup S are weakly prime if and only if they are idempotent, equivalently, if A ∩ B = (AB] for all ideals A,B of S. The ideals of an ordered semigroup S are prime if and only if they form a chain and S is intra-regular. These results have been examined already in case of both sided ordered Γ-semigroups considering the first definition of a Γ-semigroup introduced by Sen in 1981. In the present paper we keep the second definition of a Γ-semigroup introduced by Sen and Saha in 1986 (which is more general than the first one) and show that the results on ordered semigroups mentioned above can be extended to ordered Γ-semigroups without any additional conditions.

A note on the approximate amenability of semigroup algebras

Abstract: It is known that the bicyclic semigroup S1 is an amenable inverse semigroup. In this note we show that the convolution semigroup algebra l1(S1) is not approximately amenable.

First order cohomology of l1-munn algebras and certain semigroup algebras

Abstract: We characterize cyclic and weakly amenable ' 1 -Munn algebras. In the special case of Rees matrix semigroups, we obtain a new proof of the following result due to Blackmore: The semigroup algebra of every Rees matrix semigroup is weakly amenable. Char- acterizations of Connes-amenable ' 1 -Munn algebras with square sandwich matrix and semigroup algebras of Rees matrix semigroups are also provided.

A characterization of the inverse monoid of bi-congruences of certain algebras

Abstract: This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramsey's theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.

On the Type and the Minimal Presentation of Certain Numerical Semigroups

Abstract: We study numerical semigroups generated by generalized arithmetic sequences. We present a membership criterion for such a numerical semigroup, and by this we are able to answer fundamental questions concerning a numerical semigroup such as computing the Frobenius number and the type of the numerical semigroup, and decide whether the numercial semigroup is symmetric. Also for this kind of numerical semigroups, we compute the cardinality of a minimal presentation and determine whether they are complete intersections.

Ends for monoids and semigroups

Abstract: We give a graph-theoretic definition for the number of ends of Cayley digraphs for finitely generated semigroups and monoids. For semigroups and monoids, left Cayley digraphs can be very different from right Cayley digraphs. In either case, the number of ends for the Cayley digraph does not depend upon which finite set of generators is used for the semigroup or monoid. For natural numbers m and n, we exhibit finitely generated monoids for which the left Cayley digraphs have m ends while the right Cayley digraphs have n ends. For direct products and for many other semidirect products of a pair of finitely generated infinite monoids, the right Cayley digraph of the semidirect product has only one end. A finitely generated subsemigroup of a free semigroup has either one end or else has infinitely many ends.

Archimedean ordered semigroups as ideal extensions

Abstract: The main result of the paper is a structure theorem concerning the ideal extensions of archimedean ordered semigroups We prove that an archimedean ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup Conversely, if an ordered semigroup S is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then S is archimedean As a consequence, an ordered semigroup is archimedean and contains an idempotent if and only if it is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup

Filters and semigroup compactification properties

Abstract: Stone-Cech compactifications derived from a discrete semigroup S can be considered as the spectrum of the algebra ℬ(S) or as a collection of ultrafilters on S. What is certain and indisputable is the fact that filters play an important role in the study of Stone-Cech compactifications derived from a discrete semigroup.

Pseudo-amenability of Brandt semigroup algebras

Abstract: In this paper it is shown that for a Brandt semigroup $S$ over a group $G$ with an arbitrary index set $I$, if $G$ is amenable, then the Banach semigroup algebra $\ell^1(S)$ is pseudo-amenable.

Dense inverse subsemigroups of a topological inverse semigroup

Abstract: In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B ( −∞, ∞ ) 1 , a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}.