布尔矩阵的块关系矩阵及其指标格的容差关系

首先提出了布尔矩阵的块关系矩阵的定义,然后证明了布尔矩阵的所有块关系矩阵形成的格同构于该布尔矩阵指标格上的所有容差关系形成的格.     

布尔矩阵的指标格

本文首先构造了由一个布尔矩阵的特定行指标和列指标对所确定的指标格,然后刻画了指标格的同态像、直积和子格所对应的布尔矩阵的性质.     

布尔矩阵的指标格的性质

介绍了布尔矩阵的行零元、列零元和相容子矩阵的定义并讨论了它们的性质,给出了布尔矩阵的指标格分别为分配格、半分配格和半模格的等价条件.     

广义模糊幂零矩阵

路代数是加法幂等的半环,它包括了布尔代数,模糊代数,分配格及斜坡.因此布尔矩阵,模糊矩阵,格矩阵及斜矩阵都是路代数上的典型矩阵.广义模糊幂零矩阵指的就是路代数上的幂零矩阵.在2010年,Tan研究了路代数上矩阵的幂零性.在Tan的基础上继续讨论了路代数上幂零矩阵的幂零指数.     

不可约与几乎可约布尔矩阵的幂敛指数

§1.引言 布尔矩阵是指元素按如下规则运算的(0,1)矩阵:a+b=max{a,b},a·b=min{a,b}(a,b∈{0,1}),n阶布尔方阵的集合记为B_n。一个布尔方阵A的幂敛指数k(A)是满足如下条件的最小非负整数k: 条件:存在正整数p,使A~k=A~(k+p), (1.1)而称满足条件A~(k(A))=A~(k(A)+p)的最小正整数p为A的周期,记作p(A)。 对布尔矩阵的幂序列及幂敛指数的研究在有限自动机理论、二元关系理论及遍历指     

布尔矩阵平方根问题及其与图着色问题的关系

讨论了布尔矩阵平方根问题及其与图着色问题的关系.首先得到有平方根的布尔矩阵具有的一些性质;然后给出布尔矩阵存在平方根的一个充要条件;最后证明布尔矩阵的平方根问题可以转化为简单图的着色问题.     

布尔矩阵的可实现问题及其与色数问题的关系

讨论了布尔矩阵的可实现问题及其与色数问题的关系.首先给出布尔矩阵可实现的一些充要条件,讨论可实现布尔矩阵的性质,其次证明可实现布尔矩阵的容度等于该矩阵所生成的图的色数;简单图的邻接矩阵的对偶阵是可实现的,且其容度就是简单图的色数的一个上界.     

布尔矩阵的振荡条件与振荡指数

首先将常用类型的模糊矩阵都纳入到了二阶占优模糊矩阵的统一框架之内,然后利用模糊矩阵的有向伴随图,指出了强连通布尔矩阵振荡的一个充要条件,依次证明了强连通的二阶占优布尔矩阵的振荡指数为2n-2,非强连通的二阶占优布尔矩阵的振荡指数为3n-4.     

布尔网络的稳定性与镇定

研究了布尔网络的稳定性及布尔控制网络的镇定问题.利用矩阵的半张量积方法以及逻辑的矩阵表达,将布尔网络表示成离散时间动态系统,并转化成代数形式.对其代数形式的结构矩阵,建立其与一个数字变换的一种一一对应关系,再利用数字变换的方法,得出布尔网络以及布尔控制网络稳定的充要条件.     

幂等布尔矩阵的性质及构造研究

本文讨论了幂等布尔矩阵的性质及构造。首先,在研究布尔矩阵性质的基础上,给出了布尔矩阵幂等的充要条件和基于集合并运算的布尔矩阵平方的计算方法。其次,证明了(i,X)型极大传递矩阵是幂等的。最后,给出了自反幂等布尔矩阵的构造方法。     

On solvability of some types of matrix equations over lattices

We present a formula for calculating the greatest solution to matrix equations over Boolean lattices and establish the smallest (greatest) solution to a matrix equation over a distributive lattice whose matrix enjoys certain properties.     

具有有限周期的无限布尔方阵的传递指数

给出了无限布尔方阵周期的概念,研究了无限布尔方阵伴随有向图的若干性质,研究了有限布尔矩阵幂的图论性质,最后给出了无限布尔方阵传递指数的上、下界估计.     

The automorphism group of the universa distributive lattice

We describe the normal subgroup lattice of the automorphism groups of the countable universal homogeneous distributive lattice and of the countable atomless generalized Boolean lattice. Also, we show that subgroups of these automorphism groups of index less than lie between the pointwise and the setwise stabilizer of a finite set. Received March 8, 1999; accepted in final form September 16, 1999.     

Topological representation of posets

There is a canonical imbedding of a poset into a complete Boolean lattice and hence into a Boolean lattice. This gives it a representation as a collection of clopen sets of a Boolean space. There are reflective functions from a category of distributive posets to the subcategories of distributive and Boolean lattices and consequently a topological dual equivalence that extends the Stone duality of Boolean lattices.Presented by B. Jonsson.     

The primitive Boolean matrices with the second largest scrambling index by Boolean rank

The scrambling index of an n × n primitive Boolean matrix A is the smallest positive integer k such that A ^k (A ^T) ^k = J, where A ^T denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M = AB for some m × b Boolean matrix A and b×n Boolean matrix B. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M), and they also characterized all primitive matrices that achieve the upper bound. In this paper, we characterize primitive Boolean matrices that achieve the second largest scrambling index in terms of their Boolean rank.     

具有g个零强分支的有向图的传递指数

研究了传递矩阵的图论,及布尔矩阵幂的若干图论性质,给出了有向图(布尔矩阵)传递指数的上、下界估计,从而改进了已有的结果.     

Semigroups with Boolean congruence lattices

A construction of all globally idempotent semigroups with Boolean (complemented modular, relatively complemented, sectionally complemented, respectively) congruence lattice is given. Furthermore, it is shown that an arbitrary semigroup has Boolean (...) congruence lattice if and only if it is a special kind of inflation of a semigroup of the foregoing type. As applications, all commutative, finite, and completely semisimple semigroups, respectively, with Boolean (...) congruence lattice are completely determined.     

Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

In a canonical way, we establish an AZ-identity (see [2]) and its consequences, the LYM-inequality and the Sperner property, for the Boolean interval lattice. Furthermore, the Bollobas inequality for the Boolean interval lattice turns out to be just the LYM-inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice.Perhaps more surprising is that by our approach the conjecture of P. L. Erdöset al.[7] and Z. Füredi concerning an Erdös–Ko–Rado-type intersection property for the poset of Boolean chains could also be established. In fact, we give two seemingly elegant proofs.     

圈布尔矩阵类的幂敛指数集

在布尔运算下, 布尔矩阵A的幂敛指数和周期分别是使A^k=A^k+p成立的最小非负整数k和最小正整数p. 人们对周期的认识已经相当完善.给定满足一个不等式的正整数n和s, 利用组合分析确定了有向图含至少一个s -圈的n×n布尔矩阵的幂敛指数可以取得的数值.