Showing papers on "Cancellative semigroup published in 2009"
TL;DR: In this paper, the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier by the first author are investigated and the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof.
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.
40 citations
TL;DR: In this article, a large part of the paper is devoted to determining when the Bergman property can be passed from one semigroup to another and vice versa, including the notion of strong co-finality.
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.
34 citations
TL;DR: In this paper, the authors characterize when a Cayley automata semigroup is finite, is free, is a left zero semigroup, a right zero semigroup, or is a group.
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.
21 citations
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TL;DR: In this paper, the authors 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.
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.
20 citations
TL;DR: In this paper, the authors show that the twisted semigroup algebra of a regular semigroup is cellular of type J H with respect to an involution on the twisted semiigroup algebra if and only if the twisted group algebras of certain maximal subgroups are cellular.
18 citations
TL;DR: In this article, it was shown that A−1 generates an O(1 + τ) A(1 − A)−1-regularized strongly continuous semigroup on a Banach space.
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).
16 citations
TL;DR: In this paper, Carasso and Kato derived sufficient conditions for a Bernstein function to have a quasibounded holomorphic semigroup for every generator of a bounded C0-semigroup in a Banach space.
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.
16 citations
TL;DR: In this paper, it was shown 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).
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
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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.
15 citations
TL;DR: In this paper, it was shown that if S is a finite aperiodic semigroup, then the semigroup generated by the functions {φs}s ∈ S is finite.
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.
14 citations
01 Nov 2009
TL;DR: It is proved that all finitely generated commutative semigroups are FA-presentable and it is given a complete classification of the finitelygenerated FA- presentable cancellative semIGroups.
Abstract: This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups.
14 citations
01 Oct 2009
TL;DR: In this paper, a combinatorial Gelfand model for semigroup algebras of some finite semigroups was proposed, which includes the symmetric inverse semigroup, the dual symmetric inverted semigroup and the maximal factorizable subsemigroup in the dual semigroup.
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.
TL;DR: In this article, it was shown that the convolution semigroup algebra l1(S1) is not approximately amenable to the inverse semigroup S1 (S1).
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.
TL;DR: In this article, a membership criterion for numerical semigroups generated by generalized arithmetic sequences is presented, and fundamental questions concerning a numerical semigroup such as computing the Frobenius number and determining whether the numercial semigroup is symmetric.
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.
TL;DR: The main result of as discussed by the authors is a structure theorem concerning the ideal extensions of archimedean ordered semigroups, and it is shown that an ordered semigroup which contains an idempotent is an ideal extension of a simple ordered semiigroup containing an ideme-potent by a nil ordered semgroup.
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
TL;DR: In this article, the set of all doubles of a numerical semigroup is characterized, where the quotient S p = {x ∈ N ∣ p x ∈ S } is a double of S 2.
TL;DR: In this paper, it is shown that filters play an important role in the study of Stone-Cech compactifications derived from a discrete semigroup, which can be considered as the spectrum of the algebra ℬ(S) or as a collection of ultrafilters on S.
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.
TL;DR: In this paper, the set of all numerical semigroups with Frobenius number g is defined, and the generators of this semigroup are studied, where g is a positive integer.
01 Jan 2009
TL;DR: In this paper, it was shown that for a Brandt semigroup over a group with an arbitrary index set, if the group is amenable, then the Banach semigroup algebra is pseudo-amenable.
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.
TL;DR: In this article, the authors describe the equational theory of the class of cancellative entropic algebras of a fixed type and prove that a cancellative algebra embeds into an entropically polyquasigroup, a natural generalization of a quasigroup.
Abstract: We describe the equational theory of the class of cancellative entropic algebras of a fixed type. We prove that a cancellative entropic algebra embeds into an entropic polyquasigroup, a natural generalization of a quasigroup. In fact our results are even more general and some corollaries hold also for non-entropic algebras. For instance an algebra with a binary cancellative term operation, which is a homomorphism, is quasi-affine. This gives a strengthening of K. Kearnes’ theorem. Our results generalize theorems obtained earlier by M. Sholander and by J. Ježek and T. Kepka in the case of groupoids.
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TL;DR: In this paper, it was shown that almost all cancellative triple systems with vertex set [n] are tripartite, which sharpens a theorem of Nagle and Rodl on the number of cancellative three systems.
Abstract: A triple system is cancellative if no three of its distinct edges satisfy $A \cup B=A \cup C$. It is tripartite if it has a vertex partition into three parts such that every edge has exactly one point in each part. It is easy to see that every tripartite triple system is cancellative. We prove that almost all cancellative triple systems with vertex set [n] are tripartite. This sharpens a theorem of Nagle and Rodl on the number of cancellative triple systems. It also extends recent work of Person and Schacht who proved a similar result for triple systems without the Fano configuration. Our proof uses the hypergraph regularity lemma of Frankl and Rodl, and a stability theorem for cancellative triple systems due to Keevash and the second author.
TL;DR: In this article, a topological inverse semigroup from a semilattice is constructed, where the real numbers are obtained by starting with real numbers as a semi-attice with the operation a.............. ∨ b=sup{a,b}.
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}.
01 Mar 2009
TL;DR: In this article, the authors investigated some further properties of one-dimensional tiling semigroups as a particular case of the inverse semigroup associated with a factorial language and obtained a presentation for the semigroup and its description as a P*-semigroup.
Abstract: In this article, we investigate some further properties of one-dimensional tiling semigroups as a particular case of the inverse semigroup associated with a factorial language. Namely, a presentation for the semigroup and its description as a P*-semigroup are obtained. Since both cons-truc-tions rely on the language, these properties highlight the deep connection between the semigroup and the language associated with a one-dimensional tiling semigroup.
TL;DR: In this paper, the authors obtain necessary and sufficient conditions for an ordered E-inversive semigroup to be a Dubreil-Jacotin semigroup, and they also determine when such a semigroup is naturally ordered.
Abstract: Using group congruences, we obtain necessary and sufficient conditions for an ordered E-inversive semigroup to be a Dubreil-Jacotin semigroup. We also determine when such a semigroup is naturally ordered. In particular, when the subset of regular elements is a subsemigroup it contains a multiplicative inverse transversal.
TL;DR: In this paper, the authors use Mihailescu's Theorem (formerly Catalan's Conjecture) to characterize ring semigroups whose subsemigroups containing 0 form a chain with respect to set inclusion.
Abstract: A multiplicative semigroup S is called a ring semigroup if an addition may be defined on S so that (S,+,⋅) is a ring. Such semigroups have been well-studied in the literature (see Bell in Words, Languages and Combinatorics, pp. 24–31, World Scientific, Singapore, 1994; Jones in Semigroup Forum 47(1):1–6, 1993; Jones and Ligh in Semigroup Forum 17(2):163–173, 1979). In this note, we use Mihailescu’s Theorem (formerly Catalan’s Conjecture) to characterize the ring semigroups whose subsemigroups containing 0 form a chain with respect to set inclusion.
TL;DR: The zero-divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero divisors of the semigroup, with two distinct vertices joined by an edge in case their product in the semiigroup is zero as mentioned in this paper.
Abstract: The zero-divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices joined by an edge in case their product in the semigroup is zero. In this paper, we study commutative zero-divisor semigroups determined by graphs. We determine all corresponding zero-divisor semigroups of all simple graphs with at most four vertices.
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TL;DR: In this paper, it was shown that the semigroup algebra of a commutative ring with unit inverse semigroups can be described as a convolution algebra of functions on the universal \'etale groupoid associated to the inverse semigroup.
Abstract: Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality.
Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup $S$ that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent.
TL;DR: In this paper, the rank of a commutative cancellative semigroup S is the cardinality of a maximal independent subset of S. In another communication, we have characterized the structure semilattice of semigroups of rank 1 and 2.
Abstract: The rank of a commutative cancellative semigroup S is the cardinality of a maximal independent subset of S. In another communication we have characterized commutative cancellative semigroups of finite rank in several ways. We classify here semigroups of rank 1 and 2 by characterizing them in particular according to their type, which consists of their structure semilattice whose vertices are labelled by the rank of the corresponding component and distinguishing monoids from the idempotent-free ones.
TL;DR: In this article, the authors characterize q*-bisimpleur semigroups by using some kind of generalized Bruck-Reilly extensions, and show that some results concerning *-bis-imple type-Aω-semigroups given by Asibong-Ibe (Semigroup Forum 31:99-117, 1985) are generalized.
Abstract: A semigroup is called type-E if the band of its idempotents can be expressed as a direct product of a rectangular band and an ω-chain. For brevity, we call an IC *-bisimple quasi-adequate semigroup of type-E a q*-bisimple IC semigroup of type-E. In this paper, we characterize q*-bisimple semigroups by using some kind of generalized Bruck-Reilly extensions. As a consequence, some results concerning *-bisimple type-Aω-semigroups given by Asibong-Ibe (Semigroup Forum 31:99–117, 1985) are generalized.
TL;DR: In this article, the authors characterize the ring semigroups S with the property that every two nonzero subsemigroups intersect and show that every subgroup is a ring.
Abstract: A multiplicative semigroup S is said to be a ring semigroup provided there exists an addition + on S such that (S,+,⋅) is a ring. In this note, we characterize the ring semigroups S with the property that every two nonzero subsemigroups intersect.