Continuous extension of a densely parameterized semigroup

Let be a dense sub-semigroup of ℝ₊, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over can be extended to a weakly continuous semigroup over ℝ₊. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over ℝ₊. O.M. Shalit was partially supported by the Gutwirth Fellowship.     

Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or ``mirror inner". In particular, we answer a question of B. I. Plotkin.

     

The arithmetic extensions of a numerical semigroup

Abstract

In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of arithmetic extension of a given numerical semigroup. As by-product, new explicit formulas for the Frobenius number and the genus of proportionally modular semigroups are obtained.     

Class Partition Algebras as Centralizer Algebras of Wreath Products

In this article, we study an important subalgebra of the tensor product partition algebra P _k (x)? P _k (y), denoted by P _k (x, y) and called “Class Partition Algebra.” We show that the algebra P _k (n, m) is the centralizer algebra of the wreath product S _m ? S _n . Furthermore, the algebra P _k (x, y) and the tensor product partition algebra P _k (x)? P _k (y) are subalgebras of the G-colored partition algebra P _k (x;G) and G-vertex colored partition algebra P _k (x, G) respectively, for every group G with |G|=y ≥ 2k.     

The distinguishing number of the direct product and wreath product action

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted D _G(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product G ≀_Y H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups S _m × S _n on [m] × [n].     

A Class of ∗-Simple Type $A$ $ω^2$ -Semigroups (I)

In this paper, we study ∗-simple type $A$ $ω^2$-semigroups in which$\mathcal{D}^∗ =\widetilde{D}$and $\mathcal{D}^∗|_{E_S} = \mathcal{M}_d$ by the generalized Bruck-Reilly extension and obtain its structure theorem. We also obtain a criterion for isomorphisms of two such semigroups.     

Approximate amenability of certain inverse semigroup algebras

In this article, the approximate amenability of semigroup algebra ?¹(S) is investigated, where (S) is a uniformly locally finite inverse semigroup. Indeed, we show that for a uniformly locally finite inverse semigroup (S), the notions of amenability, approximate amenability and bounded approximate amenability of ?¹(S) are equivalent. We use this to give a direct proof of the approximate amenability of ?¹(S) for a Brandt semigroup (S). Moreover, we characterize the approximate amenability of ?¹(S), where S is a uniformly locally finite band semigroup.     

Graphical complexity of products of permutation groups

In this paper we study representations of permutation groups as automorphism groups of colored graphs and supergraphs. In particular, we consider how such representations for various products of permutation groups can be obtained from representations of factors and how the degree of complexity increases in such constructions.     

A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

In 1960, Sabidussi conjectured that if a graph G is isomorphic to the lexicographic product G[G], then the wreath product of by itself is a proper subgroup of . A positive answer is provided by constructing an automorphism Ψ of G[G] which satisfies: for every vertex x of G, there is an infinite subset I(x) of V(G) such that Ψ({x}×V(G))=I(x)×V(G).     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

Embedding the bicyclic semigroup into countably compact topological semigroups

We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that a topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonoff semigroup containing a copy of C(p,q).     

A survey of tensor products and related constructions in two lectures

We survey tensor products of lattices with zero and related constructions focused on two topics: amenable lattices and box products. Received August 21, 1998; accepted in final form September 9, 1998.     

P-完全正则半群

求证具有强半格结构的完全正则半群成为P-完全正则半群的充分条件.利用半群的强半格结构以及同余的性质.完全单半群的强半格-正规群带是P-完全正则半群.矩形群的强半格正规纯正群类ONBG,左群的强半格左正规纯整群类LONBG,群的强半格Clifford半群类,矩形带的强半格正规带类NB,都具有性质P.     

The convergence of Lax-Oleinik semigroup for time-periodic Lagrangian

In this article,the convergence of so-called Lax-Oleinik semigroup is studied for time-periodic Lagrangian systems when the degree of freedom is greater than 2.Under certain conditions,we show that the Lax-Oleinik semigroup converges if the rotation vector is completely irrational.Removing such conditions,we will give another kind of convergence of the sequence Fc((x,s),(x′,s′+Tn)),the convergence of which is closely related to the Lax-Oleinik semigroup.     

Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let (ℋ _t ) _t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ ^d with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ _∞ the invariant measure for (ℋ _t ) _t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L ²(γ _∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ _t ) _t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect to the invariant measure γ _∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on L ^p (γ _∞) if 1<p<∞. The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project “HARP”.     

完备右拟富足半群

在强E-右拟富足半群上定义关系γ,得到γ的性质.利用关系γ,主要研究了一类拟富足半群一完备右拟富足半群,得出这类半群的结构定理.另外,给出这类半群的另一种刻画.     

On the topological centre problem for weighted convolution algebras and semigroup compactifications

Let be a locally compact, non-compact group (we make the non-compactness assumption, for the most part, simply to avoid trivialities). We show that under a very mild assumption on the weight function , the weighted group algebra is strongly Arens irregular in the sense of Dales and Lau; i.e., both topological centres of equal . Also, we show that the topological centre of the algebra equals the weighted measure algebra . Moreover, still in the same situation, we prove that every linear (left) -module map on is automatically bounded, and even --continuous, hence given by convolution with an element in . To this end, we derive a general factorization theorem for bounded families in the -module . Finally, using this result in the case where , we give a short proof of a theorem due to Protasov and Pym, stating that the topological centre of the semigroup is empty, where denotes the -compactification of . This sharpens an earlier result by Lau and Pym; moreover, our method of proof gives a partial answer to a problem raised by Lau and Pym in 1995.

     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

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.