紧致交换矩阵半群

通过将矩阵同时对角化或同时上三角化的方法,给出有关紧致Abel矩阵半群以及紧致Hermite矩阵半群中矩阵的特征值的一些很好的刻画,证明了由可逆的Hermite矩阵构成的紧致矩阵半群中每个矩阵的特征值都是±1,Hermite矩阵单半群相似于对角矩阵半群,紧致交换矩阵半群的谱半径不超过1,等等.     

Abel正则球与完全正则半群

本文分别讨论了关于结合环和半群的二个定理，并且由结合环的这二个定理推出了如下准则：结合环Ｒ是Ａｂｅｌ正则的，当且仅当Ｒ的每个拟理想是正则环．     

Isomorphisms of Endomorphism Semigroups of Mixed Abelian Groups

We study abelian groups whose endomorphism rings are rings with unique addition. This means that there exists a unique binary operation of addition on the endomorphism semigroup which turns it into a ring. We also solve some close problems.     

Completely Decomposable Quotient Divisible Abelian Groups with Isomorphic Endomorphism Semigroups

Mathematical Notes - Let $$\Lambda$$ be a class of Abelian groups. A group $$A\in\Lambda$$ is said to be determined by its endomorphism semigroup $$E^\star(A)$$ in the class $$\Lambda$$ if every...     

The Word Problem in the Variety of Inverse Semigroups with Abelian Covers

The variety of inverse semigroups which possess E-unitary coversover Abelian groups coincides with the Mal'cev product of thevariety of semilattices and the variety of Abelian groups,andalso with the variety generated by semidirect products of semilatticesand Abelian groups. We show that this variety (and any varietyof inverse semigroups that contains this variety) has undecidableword problem.     

On the Determinability of Mixed Abelian Groups by Their Endomorphism Semigroups

Mixed Abelian groups with isomorphic endomorphism semigroups are studied. In particular, the question of when the isomorphism of endomorphism semigroups of Abelian groups implies the isomorphism of the groups themselves is investigated.     

Sumsets in difference sets

We study some properties of sets of differences of dense sets in ℤ² and ℤ³ and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results.

Sumsets being squares

Alon, Angel, Benjamini and Lubetzky [1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B?[0,N] with |B|?(logN)^15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B?[0,N] with |B|?(logN)^9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.     

Sumsets of sparse sets

Let ?? be a constant in the interval (0, 1), and let A be an infinite set of positive integers which contains at least c ₁ x ^?? and at most c ₂ x ^?? elements in the interval [1, x] for some constants c ₂ > c ₁ > 0 independent of x and each x ?? x ₀. We prove that then the sumset A + A has more elements than A (counted up to x) by a factor ${{c\left( \sigma \right)\sqrt {\log x} } \mathord{\left/ {\vphantom {{c\left( \sigma \right)\sqrt {\log x} } {\log }}} \right. \kern-0em} {\log }}$ log x for x large enough. An example showing that this function cannot be greater than ? log x is also given. Another example shows that there is a set of positive integers A which contains at least x ^?? and at most x ^??+? elements in [1, x] such that A + A is greater than A only by a constant factor. The proof of the main result is based on an effective version of Freiman??s theorem due to Mei-Chu Chang.     

Growth of Rank 1 Valuation Semigroups

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.     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Endomorphism Semigroups and Homomorphism Groups

Mathematical Notes - Let C be an Abelian group. A class X of Abelian groups is called a CE?H-class if, for every groups A, B ∈ X, the isomorphisms E?(A) ? E?(B) and...     

Sumsets contained in infinite sets of integers

If the positive integers are partitioned into a finite number of cells, then Hindman proved that there exists an infinite set B such that all finite, nonempty sums of distinct elements of B all belong to one cell of the partition. Erdös conjectured that if A is a set of integers with positive asymptotic density, then there exist infinite sets B and C such that B + C ? A. This conjecture is still unproved. This paper contains several results on sumsets contained in finite sets of integers. For example, if A is a set of integers of positive upper density, then for any n there exist sets B and F such that B has positive upper density, F has cardinality n, and B + F ? A.     

Sumsets Containing Powers of an Integer

Let n1, and let m be an integer with m2. We show that if a subset A of the interval [0,n] satisfies that 0A and |A|>1+n/2, then mA, the set of the sum of m (not necessarily distinct) elements in A, has a power of m. This result is best possible in the case that m is odd.     

Sumsets of dense sets and sparse sets

R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B:= «a+b: a ∈ A, b ∈ B» is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A+B must be piecewise Bohr. We generalize the latter result to cases where A has Banach density 0, giving a new proof of the previous results in the process.     

Inverse Additive Problems for Minkowski Sumsets II

The Brunn–Minkowski Theorem asserts that μ _d (A+B)^1/d ≥μ _d (A)^1/d +μ _d (B)^1/d for convex bodies A,B?? ^d , where μ _d denotes the d-dimensional Lebesgue measure. It is well known that equality holds if and only if A and B are homothetic, but few characterizations of equality in other related bounds are known. Let H be a hyperplane. Bonnesen later strengthened this bound by showing $$\mu_d(A+B)\geq (M^{1/(d-1)}+N^{1/(d-1)} )^{d-1}\biggl(\frac{\mu_d(A)}{M}+\frac {\mu_d(B)}{N} \biggr),$$ where M=sup?{μ _d?1((x+H)∩A)∣x∈? ^d } and $N=\sup\{\mu_{d-1}((\mathbf{y}+H)\cap B)\mid \mathbf{y}\in \mathbb {R}^{d}\}$ . Standard compression arguments show that the above bound also holds when M=μ _d?1(π(A)) and N=μ _d?1(π(B)), where π denotes a projection of ? ^d onto H, which gives an alternative generalization of the Brunn–Minkowski bound. In this paper, we characterize the cases of equality in this latter bound, showing that equality holds if and only if A and B are obtained from a pair of homothetic convex bodies by ‘stretching’ along the direction of the projection, which is made formal in the paper. When d=2, we characterize the case of equality in the former bound as well.