Showing papers on "Cancellative semigroup published in 2010"
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02 Dec 2010TL;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
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
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
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
TL;DR: In this article, the authors studied the *-automorphisms and invariant ideals of a C*-algebra generated by a commuting family of isometries, which is a natural generalization of the Toeplitz algebra.
Abstract: A C*-algebra generated by a commuting family of isometries is a natural generalization of the Toeplitz algebra. We study the *-automorphisms and invariant ideals of the C*-algebra geerated by a semigroup.
20 citations
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
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.
16 citations
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
16 citations
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.
15 citations
TL;DR: In this article, the Cayley D-saturated property of a semigroup S with respect to any subset T⊆S is characterized, and a subgraph of Cay(S,T) isomorphic to D with all vertices in V.
Abstract: Let D be a finite graph. A semigroup S is said to be Cayley D-saturated with respect to a subset T of S if, for all infinite subsets V of S, there exists a subgraph of Cay(S,T) isomorphic to D with all vertices in V. The purpose of this paper is to characterize the Cayley D-saturated property of a semigroup S with respect to any subset T⊆S. In particular, the Cayley D-saturated property of a semigroup S with respect to any subsemigroup T is characterized.
14 citations
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.
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.
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 ).
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.
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.
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...
TL;DR: In this paper, it was shown that if a Borel probability measure on the circle group is invariant under the action of a large multiplicative semigroup (lower logarithmic density is positive), then the measure is either Lebesgue or has finite support.
Abstract: We prove that if a Borel probability measure on the circle group is invariant under the action of a �large� multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then the measure is either Lebesgue or has
finite support
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).
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).
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.
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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.
01 Jan 2010
TL;DR: In this paper, it was shown that a numerical semigroup S is the quotient of infinitely many symmetric (pseudo-symmetric) numerical semigroups T by k if and only if S is itself of maximal embedding dimension.
Abstract: Let S and T be numerical semigroups and let k be a positive integer. We say that S is the quotient of T by k if an integer x belongs to S if and only if kx belongs to T . Given any integer k larger than 1 (resp., larger than 2), every numerical semigroup S is the quotient T/k of infinitely many symmetric (resp., pseudo-symmetric) numerical semigroups T by k. Related examples, probabilistic results, and applications to ring theory are shown. Given an arbitrary positive integer k, it is not true in general that every numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by k. In fact, a numerical semigroup S is the quotient of infinitely many numerical semigroups of maximal embedding dimension by each positive integer k larger than 1 if and only if S is itself of maximal embedding dimension. Nevertheless, for each numerical semigroup S, for all sufficiently large positive integers k, S is the quotient of a numerical semigroup of maximal embedding dimension by k. Related results and examples are also given.
01 Jan 2010
TL;DR: In this article, the authors considered the semigroups under composition of all linear transformations of a finite-dimensional vector space over a finite field and determined when their variants are isomorphic.
Abstract: If S is a semigroup and a ∈ S, the semigroup (S, ◦) defined by x ◦y = xay for all x,y ∈ S is called a variant of S and (S, ◦) is denoted by (S,a). In 2003-2004, Tsyaputa characterized when two variants of the following transformation semigroups are isomorphic : the symmetric inverse semigroup, the full transformation semigroup and the partial transformation semigroup on a finite nonempty set. In this paper, we consider the semigroups under composition of all linear transformations of a finite-dimensional vector space over a finite field. We determine when its variants are isomorphic. We also obtain as a consequence in the same matter for the full n × n matrix semigroup over a finite field. Mathematics Subject Classification: 20M20, 20M10
01 Jan 2010
TL;DR: In this article, the authors proposed a quasi-ideal solution for transversal transversals in semigroup, which is based on the concept of quasi-identity.
Abstract: 在这份报纸，我们与伪理想关于丰富的 semigroup 讨论一些性质足够的 transversal.Moreover，我们证明满足一些的丰富的 semigroup 的足够的 transversals 调节的二伪理想的产品是 quasiideal 足够横过。
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TL;DR: In this article, the authors consider semigroups of transformations of cellular automata which act on a fixed shift space and show that the semigroup has the ID property if the only infinite invariant closed set (with respect to the semiigroup action) is the entire space.
Abstract: In this article we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to "largeness". The first property is ID and the other property is maximal commutativity (MC). A semigroup has the ID property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space. We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring of integers mod n). It will be shown that the two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible) the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one sided shift space a semigroup acting on a two sided shift space, and vice versa, in such a way that preserves the ID and the MC properties.
TL;DR: In this article, the authors studied the semigroup OM(p,q) and OE(p and q) of all linear transformations of a vector space, and showed that OM (p, q) is a right ideal and OQ(p, q) a left ideal of T(V), respectively.
Abstract: Suppose V is an infinite-dimensional vector space and let T(V) denote the semigroup (under composition) of all linear transformations of V. In this paper, we study the semigroup OM(p,q) consisting of all α ∈ T(V) for which dim ker α ≥ q and the semigroup OE(p,q) of all α ∈ T(V) for which codim ran α ≥ q, where dimV = p ≥ q ≥ ℵ0. It is not difficult to see that OM(p,q) and OE(p,q) are a right ideal and a left ideal of T(V), respectively, and using these facts, we show that they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Also, we describe Green's relations and the two-sided ideals of each semigroup, and determine its maximal regular subsemigroup. Finally, we determine some maximal right cancellative subsemigroups of OE(p,q).
TL;DR: A semigroup S with a subband B is called a good B-quasi-Ehresmann semigroup if it is a B-semiabundant semigroup satisfying the congruence condition such that aB(a*)B(b+)b+ ⊆ B((ab)+)abB((ab*))((ab)*) for all a, b ∈ S as discussed by the authors.
Abstract: A semigroup S with a sub-band B is called a good B-quasi-Ehresmann semigroup if it is a B-semiabundant semigroup satisfying the congruence condition such that aB(a*)B(b+)b+ ⊆ B((ab)+)abB((ab)*) for all a, b ∈ S. We show that every good B-quasi-Ehresmann semigroup has a global representation and a standard representation. As a special case, the structure of good quasi-adequate semigroups is described.
01 Jan 2010
TL;DR: In this article, the authors consider the semigroup presentations of natural numbers with two and three different initial generators and union of these presentations by adding the relations, and determine the class of diagram groups over mentioned semigroup presentation, are isomorphic to
Abstract: In this paper, we consider the semigroup presentations of natural numbers with two and three different initial generators and union of these semigroup presentations by adding the relations. We will determine the class of diagram groups over mentioned semigroup presentations, are isomorphic to
01 Jan 2010
TL;DR: In this paper, it was shown that the restricted semigroup algebra r(S) is amenable as a Banach algebra if and only if S is constructed from amenable groups, without the restriction of S being E-unitary.
Abstract: In 1972, B.E. Johnson proved that for every discrete group G, (G) is amenable as a Banach algebra if and only if G is amenable as a group [9]. When S is a commutative semigroup, (S) is amenable if and only if S is a finite semilattice of abelian (and hence amenable) groups [7]. When S is a cancellative semigroup with identity, it is amenable if and only if S is an amenable group [6]. In 1978, J. Duncan and I. Namioka showed that if S is an arbitrary inverse semigroup with finite set of idempotents E(S), then (S) is amenable if and only if each maximal group of S is amenable [4]. Also, they showed that (S) fails to be amenable if E(S) is infinite, for Eunitary semigroups. In 1990, J. Duncan and A.L.T. Paterson completed the story for inverse semigroup by showing that the above result holds without the restriction of S being E-unitary [5]. Recently, G.K. Dales, A.T.-M. Lau and D. Strauss have shown that for an arbitrary semigroup S, (S) is amenable if and only if S is ’built up from amenable groups’ [3, Theorem 10.12]. They used the methods of [4]. We apply some of the above results to the semigroup algebra (Sr), where Sr is the restricted 0-semigroup associated to an inverse semigroup S [1], to prove similar results about the restricted semigroup algebra r(S), introduced by the second author and A.R. Medghalchi in [1]. We show that the restricted