Showing papers on "Bicyclic semigroup published in 2012"
TL;DR: In this article, reduced and full semigroup C Ω-algebras for left cancellative semigroups are constructed for rings of integers in number fields, and the amenability of semigroup can be expressed in terms of these semigroup ℓ-alges.
145 citations
TL;DR: In this article, the authors employ the techniques developed in an earlier paper to show that some partition semigroups arising in various contexts do not have a finite basis for their identities.
57 citations
TL;DR: In this paper, the authors studied the idempotent generated subsemigroup of the partition monoid and deduced Howie's description from 1966 of the semigroup generated by the Idempotents of a full transformation semigroup.
53 citations
TL;DR: In this paper, fixed point properties for semitopological semigroups of continuous mappings on a compact convex subset of a separated locally convex space were studied, including the class of extremely left amenable semigroup, the free commutative semigroup on one generator and the bicyclic semigroup.
43 citations
TL;DR: In this paper, affine semigroups having one Betti element are characterized and some relevant non-unique factorization invariants for these semigroup are computed for numerical semiggroups.
Abstract: We characterize affine semigroups having one Betti element and we compute some relevant non-unique factorization invariants for these semigroups As an example, we particularize our description to numerical semigroups
34 citations
TL;DR: In this paper, it was shown that the tight C*-algebra of an aperiodic tiling is the natural associative algebra to go along with an inverse semigroup.
Abstract: We realize Kellendonk’s C*-algebra of an aperiodic tiling as the tight C*-algebra of the inverse semigroup associated to the tiling, thus providing further evidence that the tight C*-algebra is a good candidate to be the natural associative algebra to go along with an inverse semigroup.
32 citations
TL;DR: In this article, the notion of restriction semigroup is generalized to P -restriction semigroup, derived instead from reducts of regular ∗-semigroups (semigroup with a regular involution), which leads naturally to dual and two-sided versions of the restriction property.
28 citations
TL;DR: The smallest monoid containing a 2-testable semigroup is defined to be a 2 testable monoid as mentioned in this paper, and the smallest semigroup can be represented by a 2 -testable monoidal semigroup.
Abstract: The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.
27 citations
13 Sep 2012
TL;DR: The paper begins with an extremely simple model of diagrams and proceeds through a series of incremental variations, all related somehow to the central theme of monoids, to illustrate the power of compositional semantics.
Abstract: The monoid is a humble algebraic structure, at first glance even downright boring. However, there's much more to monoids than meets the eye. Using examples taken from the diagrams vector graphics framework as a case study, I demonstrate the power and beauty of monoids for library design. The paper begins with an extremely simple model of diagrams and proceeds through a series of incremental variations, all related somehow to the central theme of monoids. Along the way, I illustrate the power of compositional semantics; why you should also pay attention to the monoid's even humbler cousin, the semigroup; monoid homomorphisms; and monoid actions.
25 citations
TL;DR: The double Catalan monoid as discussed by the authors is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations, and it provides an algebraization of the (combinatorial) set of 4321-avoiding permutations.
Abstract: In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.
24 citations
TL;DR: In this paper, it was shown that semigroup algebras of finite ample semigroups have generalized triangular matrix representations, and the structure of the radicals of the semigroup of such semigroup is determined.
Abstract: An adequate semigroup S is called ample if ea = a(ea)* and ae = (ae)†a for all a ∈ S and e ∈ E(S). Inverse semigroups are exactly those ample semigroups that are regular. After obtaining some characterizations of finite ample semigroups, it is proved that semigroup algebras of finite ample semigroups have generalized triangular matrix representations. As applications, the structure of the radicals of semigroup algebras of finite ample semigroups is obtained. In particular, it is determined when semigroup algebras of finite ample semigroup are semiprimitive.
Posted Content•
TL;DR: In this paper, the full and reduced C*-algebras of the left inverse hull of a semigroup were shown to be isomorphic, and conditions ensuring that these are isomorphic.
Abstract: To each discrete left cancellative semigroup $S$ one may associate a certain inverse semigroup $I_l(S)$, often called the left inverse hull of $S$. We show how the full and the reduced C*-algebras of $I_l(S)$ are related to the full and reduced semigroup C*-algebras for $S$ recently introduced by Xin Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.
TL;DR: In this article, the authors generalize this result to a sufficient condition for the non-finite basis property of semigroups, and show that a certain semigroup L of order six is not a limit variety.
Abstract: Recently, Zhang and Luo proved that a certain semigroup L of order six is non-finitely based. The main aim of the present article is to generalize this result to a sufficient condition for the non-finite basis property of semigroups. It follows that the semigroup L is inherently non-finitely based relative to a certain class of semigroups. It is also shown that the variety var L generated by L contains a unique maximal subvariety that is non-finitely based. Consequently, the variety var L is not a limit variety.
TL;DR: In this article, two equivalence relations are defined on an inverse semigroup and shown to coincide in the sense that these relations coincide with respect to the corresponding inverse semigroups.
Abstract: Let S be an inverse semigroup In Rezavand et al (Semigroup Forum 77:300–305, 2008) and Munn (Proc Glasgow Math Assoc 5:41–48, 1961) two equivalence relations are defined on S After describing these relations we show that they coincide
TL;DR: In this article, it was shown that for every n, there is a finite dimensional quadratic semigroup algebra with n generators and δ�Ω(n 2 + n 4 ) relations, where Ω n 2 is the first integer greater than the Golod-Shafarevich bound.
Abstract: A quadratic semigroup algebra is an algebra over a field given by the generators x
1, . . . , x
n
and a finite set of quadratic relations each of which either has the shape x
j
x
k
= 0 or the shape x
j
x
k
= x
l
x
m
. We prove that a quadratic semigroup algebra given by n generators and $${d\leq \frac{n^2+n}{4}}$$
relations is always infinite dimensional. This strengthens the Golod–Shafarevich estimate for the above class of algebras. Our main result however is that for every n, there is a finite dimensional quadratic semigroup algebra with n generators and δ
n
relations, where δ
n
is the first integer greater than $${\frac{n^2+n}{4}}$$
. That is, the above Golod–Shafarevich-type estimate for semigroup algebras is sharp.
TL;DR: In this article, the ordinarization transform on a non-zero non-gap semigroup is defined and the number of semigroups at each depth in the tree is analyzed.
TL;DR: In this article, the authors present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups and also for finite groups and semigroups.
Abstract: For a semigroup S its d-sequence is d(S) = (d1, d2, d3, . . .), where di is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.
Posted Content•
TL;DR: The K-theory of C*-algebras generated by the left regular representation of left regular semigroups satisfying certain regularity conditions was studied in this article.
Abstract: We compute the K-theory of C*-algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions Our result describes the K-theory of these semigroup C*-algebras in terms of the K-theory for the reduced group C*-algebras of certain groups which are typically easier to handle Then we apply our result to specific semigroups from algebraic number theory
Journal Article•
TL;DR: In this article, the Betti numbers of the numerical semigroup ring K[T ] were studied when S is a 3-generated semigroup or telescopic. And they were studied for 4-generated symmetric semigroups and the so-called 4-irreducible numerical semigroup.
Abstract: For any numerical semigroup S, there are infinitely many numerical symmetric semigroups T such that S = T/2 (see below for the definition of T/2) is their half. We are studying the Betti numbers of the numerical semigroup ring K[T ] when S is a 3-generated numerical semigroup or telescopic. We also consider 4-generated symmetric semigroups and the so called 4-irreducible numerical semigroups.
TL;DR: In this article, the authors investigate the r-ideal semigroup of a monoid and an ideal system on it, and provide conditions on the monoid such that the idempotents of the ideal system are trivial or π∗-stable.
Abstract: Let H be a monoid (resp. an integral domain) and r an ideal system on H. In this paper we investigate the r-ideal semigroup of H. One goal is to specify monoids such that their r-ideal semigroup possesses semigroup-theoretical properties, like almost completeness, π-regularity and completeness. Moreover if H is an integral domain and ∗ a star operation on H, then we provide conditions on H such that the idempotents of the ∗-ideal semigroup are trivial or such that H is π∗-stable.
TL;DR: In this article, the authors introduce the concepts of ordered quasi-ideals, ordered bi-ideal and ordered ternary semigroups and study the properties of these classes.
Abstract: . We introduce the concepts of ordered quasi-ideals, ordered bi-ideals in anordered ternary semigroup and study their properties. Also regular ordered ternarysemigroup is de ned and several ideal-theoretical characterizations of the regular orderedternary semigroups are furnished. 1. IntroductionThe literature of a ternary algebraic system was introduced by D. H. Lehmer[3] in 1932. He investigated certain ternary algebraic systems called triplexes whichturn out to be ternary groups. The notion of ternary semigroup was known toS.Banach. He showed by an example that ternary semigroup does not necessar-ily reduce to an ordinary semigroup. In [6] M. L. Santiago developed the theory ofternary semigroups. He focused his attention mainly to the study of regular ternarysemigroups, bi-ideals and ideals in ternary semigroups. The semigroup Z of all in-tegers under multiplication which plays a vital role in the literature of semigroup.The subset Z + of all positive integers of Z is a semigroup under multiplication.Now if we consider the subset Z of all negative integers of Z, then it is not a semi-group under multiplication. Taking these facts in mind D. H. Lehmer [3] introducedthe notion of ternary semigroup. Z is a natural example of a ternary semigroupunder the ternary multiplication. N. Kehayopulu in [5] developed the theory ofpo-semigroups. He mainly studied regular po-semigroups, ideals and bi-ideals inpo-semigroups. In 1999, Sang Keun Lee and Seong Gon Kang [4] gave charac-
TL;DR: A fast algorithm is obtained to compute the Castelnuovo–Mumford regularity of homogeneous semigroup rings and the Eisenbud–Goto conjecture is confirmed in a range of new cases.
Abstract: Let A⊆B be cancellative abelian semigroups, and let R be an integral domain. We show that the semigroup ring R[B] can be decomposed, as an R[A]-module, into a direct sum of R[A]-submodules of the quotient ring of R[A]. In the case of a finite extension of positive affine semigroup rings, we obtain an algorithm computing the decomposition. When R[A] is a polynomial ring over a field, we explain how to compute many ring-theoretic properties of R[B] in terms of this decomposition. In particular, we obtain a fast algorithm to compute the Castelnuovo–Mumford regularity of homogeneous semigroup rings. As an application we confirm the Eisenbud–Goto conjecture in a range of new cases. Our algorithms are implemented in the Macaulay2 package MonomialAlgebras.
TL;DR: Each factor semigroup of a free restriction (ample) semigroup over a Congruence contained in the least cancellative congruence is proved to be embeddable into a W-product of a semilattice by a monoid.
Abstract: Each factor semigroup of a free restriction (ample) semigroup over a congruence contained in the least cancellative congruence is proved to be embeddable into a W-product of a semilattice by a monoid. Consequently, it is established that each restriction semigroup has a proper (ample) cover embeddable into such a W-product.
Posted Content•
TL;DR: In this article, it was shown that the universal C � -algebra satisfying the Cuntz-Li relations is generated by an inverse semigroup of partial isometries.
Abstract: In this paper we show that the universal C � -algebra satisfying the Cuntz-Li relations is generated by an inverse semigroup of partial isometries. We apply Exel's theory of tight representations to this inverse semigroup. We identify the universal C � -algebra as the C � -algebra of the tight groupoid associated to the inverse semigroup.
TL;DR: It is proved that this problem is coNP-complete for the monoid of all full transformations of a 4-element set, which completes the description of the complexity of checking identities in the transformation monoids.
Abstract: We study the computational complexity of checking identities in a fixed finite monoid. We prove that this problem is coNP-complete for the monoid of all full transformations of a 4-element set. This result completes the description of the complexity of checking identities in the transformation monoids.
TL;DR: In this article, the existence of common fixed points for a generalized asymptotically nonexpansive semigroup T s : s ∈ S when S is a left reversible semitopological semigroup was proved.
Abstract: In this paper, we prove the existence of common fixed points for a generalized asymptotically nonexpansive semigroup { T s : s ∈ S } Open image in new window in CAT(0) spaces, when S is a left reversible semitopological semigroup. We also prove Δ- and strong convergence of such a semigroup when S is a right reversible semitopological semigroup. Our results improve and extend the corresponding results existing in the literature.
Posted Content•
TL;DR: In this paper, the authors study C*-algebras associated with subsemigroups of groups and characterize their nuclearity in terms of faithfulness of left regular representations and amenability of group actions.
Abstract: We study C*-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C*-algebras as C*-algebras of inverse semigroups, groupoid C*-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C*-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C*-algebras are UCT Kirchberg algebras. This leads to a unified approach to Cuntz algebras and ring C*-algebras.
TL;DR: In this article, the construction of a fundamental semigroup associated with a bipartite graph is introduced, which is a 0-direct union of idempotent generated completely 0-simple semigroups.
Abstract: In a manner similar to the construction of the fundamental group of a connected graph, this article introduces the construction of a fundamental semigroup associated with a bipartite graph. This semigroup is a 0-direct union of idempotent generated completely 0-simple semigroups. The maximal nonzero subgroups are the corresponding fundamental groups of the connected components. Adding labelled edges to the graph leads to a more general completely 0-simple semigroup. The basic properties of such semigroups are examined and they are shown to have certain universal properties as illustrated by the fact that the free completely simple semigroup on n generators and its idempotent generated subsemigroup appear as special cases.
Posted Content•
TL;DR: In this article, the authors give new criteria for determining when a tensor has torsion, and give constructive formulas for producing a module in the isomorphism class of the torsions submodule of M tensor N.
Abstract: Let R be a commutative Noetherian domain, and let M and N be finitely generated R-modules. We give new criteria for determining when M tensor N has torsion. We also give constructive formulas for producing a module in the isomorphism class of the torsion submodule of M tensor N. In some cases we determine bounds on the length and minimal number of generators of this module. We focus on the case where R is a numerical semigroup ring with the goal of making progress on the Huneke-Wiegand Conjecture.