On finitely generated idempotent semigroups

We give two short proofs of the well-known fact that every finitely generated idempotent semigroup is finite.     

Digital representation of semigroups and groups

A digital representation of a semigroup (S,⋅) is a family 〈F _t 〉 _t∈I , where I is a linearly ordered set, each F _t is a finite non-empty subset of S and every element of S is uniquely representable in the form Π _t∈H   x _t where H is a finite subset of I, each x _t ∈F _t and products are taken in increasing order of indices. (If S has an identity 1, then Π _t∈∅   x _t =1.) A strong digital representation of a group G is a digital representation of G with the additional property that for each t∈I, for some x _t ∈G and some m _t >1 in ℕ where m _t =2 if the order of x _t is infinite, while, if the order of x _t is finite, then m _t is a prime and the order of x _t is a power of m _t . We show that any free semigroup has a digital representation with each | F _t |=1 and that each Abelian group has a strong digital representation. We investigate the problem of whether all groups, or even all finite groups have strong digital representations, obtaining several partial results. Finally, we give applications to the algebra of the Stone-Čech compactification of a discrete group and the weakly almost periodic compactification of a discrete semigroup. Dedicated to Karl Heinrich Hofmann on the occasion of his 75th birthday. Stefano Ferri was partially supported by a research grant of the Faculty of Sciences of Universidad de los Andes. The support is gratefully acknowledged. Neil Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.     

Graphs and their associated inverse semigroups

Directed graphs have long been used to gain an understanding of the structure of semigroups, and recently the structure of directed graph semigroups has been investigated resulting in a characterization theorem and an analog of Frucht’s Theorem. We investigate two inverse semigroups defined over undirected graphs constructed from the notions of subgraph and vertex set induced subgraph. We characterize the structure of the semilattice of idempotents and lattice of ideals of these inverse semigroups. We prove a characterization theorem that states that every graph has a unique associated inverse semigroup up to isomorphism allowing for an algebraic restatement of the Edge Reconstruction Conjecture.     

Fixed-point property of random groups

In this paper, using the generalized version of the theory of combinatorial harmonic maps, we give a criterion for a finitely generated group Γ to have the fixed-point property for a certain class of Hadamard spaces, and prove a fixed-point theorem for random-group actions on the same class of Hadamard spaces. We also study the existence of an equivariant energy-minimizing map from a Γ-space to the limit space of a sequence of Hadamard spaces with the isometric actions of a finitely generated group Γ. As an application, we present the existence of a constant which may be regarded as a Kazhdan constant for isometric discrete-group actions on a family of Hadamard spaces.      

D n -normal semigroups of transformations

Given a subgroup G of the symmetric group S _n on n letters, a semigroup S of transformations of X _n is G-normal if G _S =G, where G _S consists of all permutations h∈S _n such that h ⁻¹ fh∈S for all f∈S. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups. In 1996, I. Levi showed that the alternating group A _n can not serve as the group G _S for any semigroup of total transformations of X _n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A _n -normal semigroups of partial transformations of X _n . Also, in 1994, I. Levi and R.B. McFadden described all S _n -normal semigroups. In this paper, we show that the dihedral group D _n may serve as the group G _S for semigroups of transformations of X _n . We characterize a large class of D _n -normax semigroups and describe certain D _n -normal semigroups.     

Testing polycyclicity of finitely generated rational matrix groups

We describe algorithms for testing polycyclicity and nilpotency for finitely generated subgroups of and thus we show that these properties are decidable. Variations of our algorithm can be used for testing virtual polycyclicity and virtual nilpotency for finitely generated subgroups of .

     

圈积的词长度公式

利用Fordham的技巧证明了圈积群HιZ的一个词长度公式,其中H是一个有限生成群,Z是整数群.     

On Some Properties of the Quaternionic Functional Calculus

In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic Banach space. That functional calculus is based on the theory of slice regular functions and on a Cauchy formula which holds for particular domains where the admissible functions have power series expansions. In this paper, we use a new version of the Cauchy formula with slice regular kernel to extend the validity of the quaternionic functional calculus to functions defined on more general domains. Moreover, we show some of the algebraic properties of the quaternionic functional calculus such as the S-spectral radius theorem and the S-spectral mapping theorem. Our functional calculus is also a natural tool to define the semigroup e ^tA when A is a linear quaternionic operator.      

Algebraic K-theory of generalized free products and functors Nil

In this paper, we extend Waldhausen's results on algebraic K-theory of generalized free products in a more general setting and we give some properties of the Nil functors. As a consequence, we get new groups with trivial Whitehead groups.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

The Mal'cev Rank for Modules over Pseudo-Valuation Domains

Let R be a commutative ring and A be an R-module. The Mal'cev rank μ(A) of A is the sup of genN, where N ranges over the finitely generated submodules of A, and genN is the minimum number of generators of N. We prove that μ is both sub-additive and pre-additive as an invariant of Mod(R). Our main goal is to investigate μ for modules over pseudo-valuation domains. Specifically, we establish which pseudo-valuation domains R satisfy the property that an R-module of finite Mal'cev rank must be finitely generated. We split the class 𝒞 of pseudo-valuation domains as a union 𝒞 = 𝒞₁ ∪ 𝒞₂ ∪ 𝒞₃ ∪ 𝒞₄ of suitably defined subclasses, and prove that the property holds if and only if R ∈ 𝒞₃ ∪ 𝒞₄. In that case we can describe the R-modules A where μ(A) < ∞. We also show that, for R ∈ 𝒞₄, there exist indecomposable R-modules of arbitrarily large finite Mal'cev rank.     

Generation of the integrated semigroups by superelliptic differential operators

Let A be a superelliptic differential operator of order 2m introduced by E.B. Davies [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169]. In the case of 2m>N, he obtained the upper Gaussian bound of the integral kernel representing (e−zA)_z∈C+ and the estimates of the Lp-operator norm of the semigroup for all p∈[1,∞). The purpose of the present paper is to show that −i(A+k) (for some constant k>0) generates an integrated semigroup on Lα,p (weighted Lp space) and lp(Lα,q). To prove this we need norm estimates of (e−zA)_z∈C+ on each of these spaces. Also we get another norm estimate of (e−zA)_z∈C+ on Lp when 2m>N without using the integral kernel. This norm estimate is better than that in [E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169] and gives a better “times of the integration” of the integrated semigroup.     

m-Systems and Partial m-Systemsof Polar Spaces

LetP be a finite classical polar space of rankr, withr 2. A partialm-systemM ofP, with 0 m r - 1, is any set (₁), ₂,..., _k ofk ( 0) totally singularm-spaces ofP such that no maximal totally singular space containing _i has a point in common with (_{1 2} ... _k) — _i,i = 1, 2,...,k. In a previous paper an upper bound for ¦M¦ was obtained (Theorem 1). If ¦M¦ = , thenM is called anm-system ofP. Form = 0 them-systems are the ovoids ofP; form =r - 1 them-systems are the spreads ofP. In this paper we improve in many cases the upper bound for the number of elements of a partialm-system, thus proving the nonexistence of several classes ofm-systems.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Classifying pointed hopf algebras of dimension 16

We classify pointed Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero. Apart from the 11 group algebras, there are 29 such Hopf algebras. All of them can be obtained using the Ore extension construction, as described recently by Beattie, the second author, and Grunenfelder.     

Lower bounds for dimensions of representation varieties

The set of -dimensional complex representations of a finitely generated group form a complex affine variety . Suppose that is such a representation and consider the associated representation on complex matrices obtained by following with conjugation of matrices. Then it is shown that the dimension of at is at least the difference of the complex dimensions of and . It is further shown that in the latter cohomology may be replaced by various proalgebraic groups associated to and .

     

Standard triangularization of semigroups of non-negative operators

We show that, under mild conditions, a semigroup of non-negative operators on L^p(X,μ) (for 1?p<∞) of the form scalar plus compact is triangularizable via standard subspaces if and only if each operator in the semigroup is individually triangularizable via standard subspaces. Also, in the case of operators of the form identity plus trace class we show that triangularizability via standard subspaces is equivalent to the submultiplicativity of a certain function on the semigroup.