Showing papers on "Cancellative semigroup published in 2006"

Guarded and Banded Semigroups

Abstract: The variety of guarded semigroups consists of all (S,·, ¯) where (S,·) is a semigroup and x ↦ \overline{x} is a unary operation subject to four additional equations relating it to multiplication. The semigroup Pfn(X) of all partial transformations on X is a guarded semigroup if x \overline{f} = x when xf is defined and is undefined otherwise. Every guarded semigroup is a subalgebra of Pfn(X) for some X. A covering theorem of McAlister type is obtained. Free guarded semigroups are constructed paralleling Scheiblich's construction of free inverse semigroups. The variety of banded semigroups has the same signature but different equations. There is a canonical forgetful functor from guarded semigroups to banded semigroups. A semigroup underlies a banded semigroup if and only if it is a split strong semilattice of right zero semigroups. Each banded semigroup S contains a canonical subsemilattice g⋆(S). For any given semilattice L, a construction to synthesize the general banded semigroup S with g⋆ ≅ L is obtained.

Connes-amenability of bidual and weighted semigroup algebras

Abstract: We investigate the notion of Connes-amenability, introduced by Runde in [10], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced in [13], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as ${ C}^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group, and the weight satisfying a certain restrictive condition. This latter point was first shown by Gronnbaek in [6], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like ${ C}^*$-algebras.

On the importance of idempotence

Abstract: Range searching is among the most fundamental problems in computational geometry. An n-element point set in Rd is given along with an assignment of weights to these points from some commutative semigroup. Subject to a fixed space of possible range shapes, the problem is to preprocess the points so that the total semigroup sum of the points lying within a given query range η can be determined quickly. In the approximate version of the problem we assume that η is bounded, and we are given an approximation parameter e > 0. We are to determine the semigroup sum of all the points contained within η and may additionally include any of the points lying within distance e • diam(η) of η's boundar.In this paper we contrast the complexity of range searching based on semigroup properties. A semigroup (S,+) is idempotent if x + x = x for all x ∈ S, and it is integral if for all k ≥ 2, the k-fold sum x + ... + x is not equal to x. For example, (R, min) and (0,1, ∨) are both idempotent, and (N, +) is integral. To date, all upper and lower bounds hold irrespective of the semigroup. We show that semigroup properties do indeed make a difference for both exact and approximate range searching, and in the case of approximate range searching the differences are dramatic.First, we consider exact halfspace range searching. The assumption that the semigroup is integral allows us to improve the best lower bounds in the semigroup arithmetic model. For example, assuming O(n) storage in the plane and ignoring polylog factors, we provide an Ω*(n2/5) lower bound for integral semigroups, improving upon the best lower bound of Ω*(n1/3), thus closing the gap with the O(n1/2) upper bound.We also consider approximate range searching for Euclidean ball ranges. We present lower bounds and nearly matching upper bounds for idempotent semigroups. We also present lower bounds for range searching for integral semigroups, which nearly match existing upper bounds. These bounds show that the advantages afforded by idempotency can result in major improvements. In particular, assuming roughly linear space, the exponent in the e-dependencies is smaller by a factor of nearly 1/2. All our results are presented in terms of space-time tradeoffs, and our lower and upper bounds match closely throughout the entire spectrum.To our knowledge, our results provide the first proof that semigroup properties affect the computational complexity of range searching in the semigroup arithmetic model. These are the first lower bound results for any approximate geometric retrieval problems. The existence of nearly matching upper bounds, throughout the range of space-time tradeoffs, suggests that we are close to resolving the computational complexity of both idempotent and integral approximate spherical range searching in the semigroup arithmetic model.

Automorphisms of the Endomorphism Semigroup of a Free Inverse Semigroup

Abstract: We prove that automorphisms of the endomorphism semigroup of a free inverse semigroup are inner and determine all isomorphisms between the endomorphism semigroups of free inverse semigroups.

Spectral Properties of a Class of Positive Semigroups on Banach Lattices and Streaming Operators

Abstract: We deal with a general streaming semigroup arising in transport theory. We show that in general such a semigroup has a direct decomposition into three simpler semigroups one of which extends to a group. A general spectral theory of them is given and is partly derived from that of a class of positive semigroups on Banach lattices.

The semigroup of conjugates of a word

Abstract: We introduce a finite semigroup associated with a conjugacy class of a word in the free monoid over a finite alphabet. Using properties of this semigroup we derive results on combinatorics on words.

Nilpotent linear semigroups

Abstract: We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.

The structure of L ∗ -inverse semigroups

Abstract: The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products.

Lattices embeddable in subsemigroup lattices. II. Cancellative semigroups

Abstract: Repnitskii proved that any lattice embeds in a subsemigroup lattice of some commutative, cancellative, idempotent free semigroup with unique roots. In that proof, use is made of a result by Bredikhin and Schein stating that any lattice embeds in a suborder lattice of suitable partial order. Here, we present a direct proof of Repnitskii’s result which is independent of Bredikhin-Schein’s, thus giving the answer to the question posed by Shevrin and Ovsyannikov.

Unique prime factorization in a partial semigroup of matrix-polynomials

Abstract: Let K be a field. We consider divisability and factorization into irreducibel elements in the partial semigroup of 2 × 2-matrices with entries in K[x] and determinant 1, where multiplication is defined as matrix-multiplication if the degrees of the factors add up, cf. Section 2. Our aim is to establish a unique factorization result, cf. Theorem 3.1. Although our considerations are purely algebraic and in fact quite elementary, they should be seen in connection with some results of complex analysis. Let us explain this motivation: Let W (z) = (wij(z))i,j=1,2 be a 2 × 2-matrix function whose entries are entire functions, i.e. are defined and holomorphic in the whole complex plane. We say that W belongs to the class Mκ where κ is a nonnegative integer, if wij(z) = wij(z), W (0) = I, detW (z) = 1, and if the kernel KW (w, z) := W (z)JW (w) − J z − w has κ negative squares. Thereby

A group-embeddable non-automatic semigroup whose universal group is automatic

Abstract: Answering a question of Hoffmann and of Kambites, an example is exhibited of a finitely generated semigroup $S$ such that $S$ embeds in a group and $S$ is not automatic, but the universal group of $S$ is automatic.

On Distribution Semigroups with a Singularity at Zero and Bounded Solutions of Differential Inclusions

Abstract: We study distribution semigroups with a singularity at zero and their generators, and establish a relationship between this semigroup and a degenerate semigroup of linear operators on the open right half-line. The study makes an intensive use of spectral theory of linear relations. Applications to the existence problem for bounded solutions of linear differential inclusions are obtained.

Structure of a permutable Munn semigroup of finite rank

Abstract: A semigroup any two congruences of which commute as binary relations is called a permutable semigroup. We describe the structure of a permutable Munn semigroup of finite rank.

Structure of commutative cancellative subarchimedean semigroups

Abstract: A commutative semigroup $S$ is subarchimedean if there is an element $z\in S$ such that for every $a\in S$ there exist a positive integer $n$ and $x\in S$ such that $z^n=ax$. Such a semigroup is archimedean if this holds for all ${z\in S}$. A commutative cancellative idempotent-free archimedean semigroup is an $\frak{N}$-semigroup. We study the structure of semigroups in the title as related to $\frak{N}$-semigroups.

On quasi-Frobenius semigroup algebras

Abstract: We define quasi-Frobenius semigroups and find necessary and sufficient conditions under which a semigroup algebra of a 0-cancellative semigroup is quasi-Frobenius.

Commutative cancellative semigroups and rational vector spaces

Abstract: Representing a commutative cancellative subarchimedean semigroup S as Ni(G, I), we consider Hom(S, Q) and Hom(G, Q), where Q is the additive group of rational numbers. These sets can be given the structure of rational vector spaces. Suitable isomorphic copies of these vector spaces are found by means of certain functions related to some mappings introduced by T. Tamura.

Regular Semigroups Each of Whose Least Completely Simple Congruence Classes Has a Greatest Element

Abstract: A regular semigroup S is termed locally F-regular, if each class of the least completely simple congruence ξ contains a greatest element with respect to the natural partial order. It is shown that each locally F-regular semigroup S admits an embedding into a semidirect product of a band by S/ξ. Further, if ξ satisfies the additional property that for each s ∊ S and each inverse (sξ)′ of sξ in S/ξ the set (sξ)′ ∩ V(s) is not empty, we represent S both as a Rees matrix semigroup over an F-regular semigroup as well as a certain subsemigroup of a restricted semidirect product of a band by S/ξ.

Semigroup of lipschitz operators

Abstract: Lipschitzian semigroup is a semigroup of Lipschitz op- erators which contains C0 semigroup and nonlinear semigroup. In this paper, we establish the cannonical exponential formula of Lip- schitzian semigroup from its Lie generator and the approximation theorem by Laplace transform approach to Lipschitzian semigroup.

The structure of a kind of semiabundant semigroups

Abstract: In this paper we study semiabundant semigroups whose idempotents form normal subbands. We prove that a semigroup is of this type if and only if it is isomorphic to a quasi-spined product of a left normal band, a right normal band and a semiadequate semigroup.

Notes on grading monoids

Abstract: . Throughout this paper, a semigroup S will denote atorsion free grading monoid, and it is a non-zero semigroup with0. The operation is written additively. The aim of this paper isto study semigroup version of an integral domain ([1],[3],[4] and[5]). 1. IntroductionLet S be an additive commutative semigroup with identity(denotedby 0), that is a monoid. A monoid S is said to be cancellative ifx + y = x + z with x,y and z ∈ S implies y = z and S is said to betorsion-free if nx = ny with x,y ∈ S and n ∈ N implies x = y whereN denotes the set of all positive integers. A cancellative monoid iscalled a grading monoid [10,p.112]. In this paper, a semigroup S willdenote a torsion free grading monoid,and it is a non-zero semigroupwith 0. The operation is written additively.A nonempty subset B of a semigroup S is called an additive systemif it satiesfies the following condition b 1 ,b 2 ∈ B ⇒ b 1 + b 2 ∈ B. Foran additive system B, the quotient semigroup S B is defined as follows:{s − b | s ∈ S,b ∈ B}. Especially, if B = S, then the quotientsemigroup S