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Showing papers on "Bicyclic semigroup published in 2010"


Book
02 Dec 2010
TL;DR: In this article, the authors studied the structure of the second dual of the semigroup algebra and its amenability constant, showing that there are 'forbidden values' for this constant.
Abstract: Let $S$ be a (discrete) semigroup, and let $\ell^{\,1}(S)$ be the Banach algebra which is the semigroup algebra of $S$. The authors study the structure of this Banach algebra and of its second dual. The authors determine exactly when $\ell^{\,1}(S)$ is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are 'forbidden values' for this constant. Table of Contents: Introduction; Banach algebras and their second duals; Semigroups; Semigroup algebras; Stone-?ech compactifications; The semigroup $(\beta S, \Box)$; Second duals of semigroup algebras; Related spaces and compactifications; Amenability for semigroups; Amenability of semigroup algebras; Amenability and weak amenability for certain Banach algebras; Topological centres; Open problems; Bibliography; Index of terms; Index of symbols. (MEMO/205/966)

176 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C ( p, q ) and proved that a topological semiigroup S with pseudocompact square contains no dense copy of C( p, q ).

54 citations


Journal ArticleDOI
TL;DR: The partition monoid is a salient natural example of a *-regular semigroup, and it has been shown that it contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set.
Abstract: The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

47 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the monoid of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity (a.k.a., A^2B^4A^2A^A^B+A+B+C+C^A+C +B+B +A^C+A +C+B^C +A+2B+4A+A−2B +C^C^2C+4B+2A−4A +2B−4B +4A
Abstract: We show that the monoid $M_{2}(\mathbb {T})$ of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity $$A^2B^4A^2A^2B^2A^2B^4A^2=A^2B^4A^2B^2A^2A^2B^4A^2.$$ Studying reduced identities for subsemigroups of $M_{2}(\mathbb {T})$ , and introducing a faithful semigroup representation for the bicyclic monoid by 2×2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.

45 citations


Journal ArticleDOI
TL;DR: In this paper, a simpler approach to counting numerical semigroups of a given genus was proposed, without referring to the generators or the semigroup tree, and an improved asymptotic lower bound was given.
Abstract: Let ng denote the number of numerical semigroups of genus g. Bras-Amoros conjectured that ng possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree. We offer a new, simpler approach to counting numerical semigroups of a given genus. Our method gives direct constructions of families of numerical semigroups, without referring to the generators or the semigroup tree. In particular, we give an improved asymptotic lower bound for ng.

42 citations


Journal ArticleDOI
TL;DR: It is proved that the fundamental group of coincides with the maximum group image of T in terms of the universal locally constant covering of its classifying topos.
Abstract: We examine an inverse semigroup T in terms of the universal locally constant covering of its classifying topos Open image in new window. In particular, we prove that the fundamental group of Open image in new window coincides with the maximum group image of T. We explain the connection between E-unitary inverse semigroups and locally decidable toposes, characterize E-unitary inverse semigroups in terms of a kind of geometric morphism called a spread, characterize F-inverse semigroups, and interpret McAlister’s “P-theorem” in terms of the universal covering.

30 citations


Journal ArticleDOI
TL;DR: In this paper, the twirling semigroups of super operators are studied, namely certain quantum dynamical semiigroups that are associated with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group.
Abstract: We study the twirling semigroups of (super) operators, namely certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.

28 citations


Journal ArticleDOI
TL;DR: In this article, a semigroup variety is called a variety of degree ≤ 2 if all its nilsemigroups are semigroups with zero multiplication and if all semigroup varieties of degree > 2 have zero multiplication unless they are upper-modular elements of the lattice.
Abstract: A semigroup variety is called a variety of degree ≤2 if all its nilsemigroups are semigroups with zero multiplication, and a variety of degree >2 otherwise. We completely determine all semigroup varieties of degree >2 that are upper-modular elements of the lattice of all semigroup varieties and find quite a strong necessary condition for semigroup varieties of degree ≤2 to have the same property.

23 citations


Journal ArticleDOI
TL;DR: In this paper, the concept of fuzzy generalized bi-ideals of an ordered semigroup was introduced, and two main theorems which characterize regular orderedsemigroups and intra-regular ordered semigroups in terms of fuzzyleft ideals, fuzzy right ideal, fuzzy bi-id ideals or fuzzy quasi-ideal were given.
Abstract: Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper, the concept of fuzzygeneralized bi-ideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.Finally, two main theorems which characterize regular orderedsemigroups and intra-regular ordered semigroups in terms of fuzzyleft ideals, fuzzy right ideals, fuzzy bi-ideals or fuzzyquasi-ideals are given. The paper shows that one can pass fromresults in terms of fuzzy subsets in semigroups to orderedsemigroups. The corresponding results of unordered semigroups arealso obtained.

22 citations


Journal ArticleDOI
James East1
TL;DR: The partial transformation semigroup $\mathcal{PT}_n$ is the semigroup of all partial transformations on the finite set n = {1,…, n}.
Abstract: The partial transformation semigroup $\mathcal{PT}_n$ is the semigroup of all partial transformations on the finite set n = {1,…, n}. The transformation semigroup $\mathcal{T}_n\subseteq\mathcal{PT...

17 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that if a semigroup T divides into a semidirect product S⋊T where S is a finite semilattice whose natural order makes S a chain, then so does any semi-directional product S ⋊ T where T is a chain.
Abstract: We show that if a semigroup T divides a semigroup of full order preserving transformations of a finite chain, then so does any semidirect product S⋊T where S is a finite semilattice whose natural order makes S a chain.

01 Jan 2010
TL;DR: The notion of a torsor for an inverse semigroup, which is based on semigroup actions, has been introduced in this article, and it has been shown that this is precisely the structure classied by the topos associated with an inverse semiigroup.
Abstract: We dene the notion of a torsor for an inverse semigroup, which is based on semigroup actions, and prove that this is precisely the structure classied by the topos associated with an inverse semigroup Unlike in the group case, not all set-theoretic torsors are isomorphic: we shall give a complete description of the category of torsors We explain how a semigroup prehomomorphism gives rise to an adjunction between a restrictions-of-scalars functor and a tensor product functor, which we relate to the theory of covering spaces and E-unitary semigroups We also interpret for semigroups the Lawvere-product of a sheaf and distribution, and nally, we indicate how the theory might be extended to general semigroups, by dening a notion of torsor and a classifying topos for those

Journal ArticleDOI
TL;DR: In this paper, the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra, is described in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X.
Abstract: This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T,A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.

Journal ArticleDOI
TL;DR: It is proved that the full transformation monoid on a countably infinite set is isomorphic to a submonoid of End(R), the endomorphism monoid of the infinite random graph R, which has an undecidable universal theory.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the symmetric inverse topological semigroup of finite transformations of the rank ≤ n is algebraically h -closed in the class of topological inverse semigroups.
Abstract: We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.

Journal ArticleDOI
TL;DR: It is shown that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max{|S1|,|S2|}.
Abstract: It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2 whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max{|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the $\mathcal{J}$-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the $\mathcal{R}$-classes to be finite.

Journal ArticleDOI
TL;DR: In this article, the authors consider a rank 1 valuation semigroup S of a local ring R centered on R and show that the Hilbert polynomial of R gives a bound on the growth rate of S. This allows them to give a very simple example of a well ordered subsemigroup of Q+ which is not a value semigroup of local domain.
Abstract: We consider semigroup S of a rank 1 valuation ? centered on a local ring R. We show that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S. This allows us to give a very simple example of a well ordered subsemigroup of Q+, which is not a value semigroup of a local domain. We also consider the rates of growth which are possible for S. We show that quite exotic behavior can occur. In the final section, we consider the general question of characterizing rank 1 value semigroups.

Journal ArticleDOI
TL;DR: A natural extension of the usual definition of M-automata is considered which permits the automaton to utilise more of the structure of each monoid, and additionally allows it to be defined for S an arbitrary semigroup, and resulting automata are equivalent to the valence automata with rational target sets which arise in the theory of regulated rewriting systems.

Journal ArticleDOI
Paul Ramsden1
TL;DR: For weakly cancellative inverse semigroups, the injectivity of a Banach right module is studied in this paper, which is the same as studying the flatness of the predual left module c 0 (S ).

Journal ArticleDOI
TL;DR: In this paper, it was shown that T(t)-T(2t) has a norm approaching 2 near the origin, where T is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane.
Abstract: This paper is concerned first with the behaviour of differences T(t)-T(s) near the origin, where (T(t)) is a semigroup of operators on a Banach space, defined either on the positive real line or a sector in the right half-plane (in which case it is assumed analytic). For the non-quasinilpotent case extensions of results in the published literature are provided, with best possible constants; in the case of quasinilpotent semigroups on the half-plane, it is shown that, in general, differences such as T(t)-T(2t) have norm approaching 2 near the origin. The techniques given enable one to derive estimates of other functions of the generator of the semigroup; in particular, conditions are given on the derivatives near the origin to guarantee that the semigroup generates a unital algebra and has bounded generator.


Journal ArticleDOI
TL;DR: In this paper, the authors used automated reasoning to solve a number of problems regarding the axiomatic definition of inverse semigroups and provided a single identity for groups in terms of two binary operations.
Abstract: In 1981, Tamura posed a number of problems regarding the axiomatic definition of inverse semigroups. The main goal of this article is to use automated reasoning to solve these problems. In the process, we find some new defining sets of identities for the class of inverse semigroups and provide a single identity for groups in terms of two binary operations.

Journal ArticleDOI
TL;DR: In this article, the authors define the radical ϱ ≥ ≥ 0 (k ≥ 0) of a relation ϱ on an arbitrary semigroup and define various types of k-regularity of semigroups.
Abstract: In this paper we define the radical ϱ k (k∈Z +) of a relation ϱ on an arbitrary semigroup. Also, we define various types of k-regularity of semigroups and various types of k-Archimedness of semigroups. Using these notions we describe the structure of semigroups in which ρ k is a band (semilattice) congruence for some Green’s relation.

Journal ArticleDOI
TL;DR: The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products and the permutation conjugacy relation in this semigroup and the Green's rel...
Abstract: The inverse semigroup of partial automaton permutations over a finite alphabet is characterized in terms of wreath products. The permutation conjugacy relation in this semigroup and the Green's rel...

Journal ArticleDOI
TL;DR: In this paper, the zero-divisors of the semigroup module were studied and the authors showed that if $M$ is an $R$-module and $S$ is a commutative, cancellative and torsion-free monoid, then the $R[S]$]-module has few zero-Divisors if and only if
Abstract: Let $M$ be an $R$-module and $S$ a semigroup. Our goal is to discuss zero-divisors of the semigroup module $M[S]$. Particularly we show that if $M$ is an $R$-module and $S$ a commutative, cancellative and torsion-free monoid, then the $R[S]$-module $M[S]$ has few zero-divisors of degree $n$ if and only if the $R$-module $M$ has few zero-divisors of degree $n$ and Property (A).

Journal ArticleDOI
John Fountain1
TL;DR: In this article, the authors give an account of some of the highlights of the mathematical research of Douglas Munn and give an overview of the main contributions of Munn's work.
Abstract: We give an account of some of the highlights of the mathematical research of Douglas Munn.

Posted Content
TL;DR: The semigroup of as mentioned in this paper has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or groupsomorphisms.
Abstract: In this paper we study the semigroup $I_\infty^\dnearrow(N)$ of partial co-finite almost monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $I_\infty^\dnearrow(N)$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. Also we prove that every Baire topology $\tau$ on $I_\infty^\dnearrow(N)$ such that $(I_\infty^\dnearrow(N),\tau)$ is a semitopological semigroup is discrete, describe the closure of $(I_\infty^\dnearrow(N),\tau)$ in a topological semigroup and construct non-discrete Hausdorff semigroup topologies on $I_\infty^\dnearrow(N)$.

Journal ArticleDOI
TL;DR: The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B n of n − 1 zero-one matrices with the Boolean matrix product as discussed by the authors.
Abstract: The semigroup of binary relations on {1,…, n} with the relative product is isomorphic to the semigroup B n of n × n zero-one matrices with the Boolean matrix product. Over any field F, we prove that the semigroup algebra FB n contains an ideal K n of dimension (2 n − 1)2, and we construct an explicit isomorphism of K n with the matrix algebra M 2 n −1(F).

Journal ArticleDOI
TL;DR: In this paper, the authors consider the endomorphisms of a Brandt semigroup B n and the semigroup of mappings E(B n ) that they generate under pointwise composition.
Abstract: We consider the endomorphisms of a Brandt semigroup B n and the semigroup of mappings E(B n ) that they generate under pointwise composition. We describe all the elements of this semigroup, determine Green's relations, consider certain special types of mapping, which we can enumerate for each n, and give complete calculations for the size of E(B n ) for small n.

Posted Content
TL;DR: In this paper, a general theorem about the cellularity of twisted semigroup algebras of regular semigroups was proved, which generalises a recent result of East about semi-gigas of inverse semiggroups.
Abstract: The Temperley-Lieb and Brauer algebras and their cyclotomic analogues, as well as the partition algebra, are all examples of twisted semigroup algebras. We prove a general theorem about the cellularity of twisted semigroup algebras of regular semigroups. This theorem, which generalises a recent result of East about semigroup algebras of inverse semigroups, allows us to easily reproduce the cellularity of these algebras.