A Combinatorial Property and Cayley Graphs of Semigroups

Abstract. We describe semigroups satisfying a combinatorial property defined in terms of Cayley graphs.     

A Combinatorial Property and Divisibility Graphs of Monomial Matrix Semigroups

We describe a class of monomial matrix semigroups satisfying a combinatorial property defined in terms of divisibility graphs.     

矩阵逆半群

讨论矩阵逆半群的一些基本性质, 证明矩阵逆半群的幂等元集是有限布尔格的子半格, 从而证明等秩矩阵逆半群是群, 然后完全确定二级矩阵逆半群的结构：一个二级矩阵逆半群或者同构于二级线性群,或者同构于二级线性群添加一个零元素,或者是交换线性群的有限半格, 或者满足其他一些性质; 对于由某些二级矩阵构成的集合, 我们给出了它们成为矩阵逆半群的充分必要条件.     

Combinatorial Results for Semigroups of Order-Preserving Full Transformations

Let O_n be the semigroup of all order-preserving full transformations of a finite chain, say X_n = {1, 2, ..., n}, and for a given full transformation α: X_n → X_n let f(α) = |{x ∈ X_n: xα = x}|. In this note we obtain and discuss formulae for f(n,r,k) = |{α → O_n: f(α) = r ∧ max(Im α) = k}| and J(n,r,k) = |{α → O_n: |Im α| = r ∧ max(Im α) = k}|. We also obtain similar results for E(O_n), the set of idempotents of O_n.     

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.     

Hamilton算子矩阵的半群生成定理

研究了Hamilton算子矩阵的半群生成问题,得到其生成C_0半群的若干充分必要条件,并给出其成为半群无穷小生成元时的谱分布.作为应用,基于Hamilton系统的半群方法,给出一类四阶微分方程混合问题的古典解.     

Fischer Matrices for Generalised Symmetric Groups—A Combinatorial Approach

The main aim of this article is to give a combinatorial method for computing the Fischer matrices of the generalised symmetric group. It is shown that generalised Young tableaux and tabloids play a crucial role in the calculation of the Fischer matrices. In particular, a recursive method which is similar to the Murnaghan-Nakayama formula involved in the calculation of irreducible characters of the symmetric group is proved.     

生灭矩阵生成压缩半群的条件

研究了生灭矩阵本身何时在l1空间中存成压缩半群,并获得一个充要条件。     

幂零矩阵的一个性质

通过对2009年全国硕士研究生入学考试数学试卷中的一道线性代数试题的推广,得到幂零矩阵的一个性质,从而揭示出该试题的命题思路与背景.     

交换矩阵半群的可约性

首先分别给出单生矩阵半群或者摹群不可约、不可分解以及完全可约的充分必要条件,其次讨论一般域上矩阵半群的可约性的一些条件,最后特别地讨论实数域上矩阵半群的可约性,完全确定了实数域上对称和反对称矩阵组成的不可约交换矩阵半群.     

A Characterization of Markov Property for Semigroups with Invariant Measure

The positivity of the carré du champ operator is a direct consequence of the positivity of the associated Markov semigroup. We show in this note that the reciprocal implication holds in invariant measure, under minimal hypotheses of continuity and stability.     

Irreducible Semigroups of Matrices with Eigenvalue One

We study groups and semigroups of n x n matrices with the property that each matrix has a fixed point, i.e., 1 is an eigenvalue of each matrix. We show that for n=3 and $n\geq 5$ there are irreducible matrix groups and irreducible semigroups of nonnegative matrices with this property. In fact, for n = 3 we determine the structure of any such semigroup. We also present additional hypotheses implying reducibility.     

The Ideal Retraction Property for Idempotent Semigroups

A semigroup S is said to have the ideal retraction property provided each of its ideals is the image of an idempotent endomorphism of S. The main result of this work is a characterization of those bands which have the idempotent retraction property. All such bands are normal.     

广义循环矩阵的一个性质

可表为非奇异对角矩阵和循环矩阵乘积的矩阵,我们称其为广义循环矩阵.本文给出了单位矩阵与广义循环矩阵的和矩阵的非奇异的充要条件,得到了这样和矩阵的相对增益阵列的显示表达式.     

Homomorphisms Between Multiplicative Semigroups of Matrices Over Fields

Suppose F is a field, and n, p are integers with 1 ≤ p 〈 n. Let Mn（F） be the multiplicative semigroup of all n × n matrices over F, and let M^Pn（F） be its subsemigroup consisting of all matrices with rank p at most. Assume that F and R are subsemigroups of Mn（F） such that F M^Pn（F）. A map f ： F→R is called a homomorphism if f（AB） = f（A）f（B） for any A, B ∈F. In particular, f is called an endomorphism if F = R. The structure of all homomorphisms from F to R （respectively, all endomorphisms of Mn（F）） is described.     

A New Proof of the Protasov-Voynov Theorem on Semigroups of Nonnegative Matrices

A new combinatorial proof of the Protasov-Voynov theorem on the structure of irreducible semigroups of nonnegative matrices is proposed. The original proof was obtained by geometric methods.