Congruence numerical ranges and their radii

The earliest congruence numerical range was introduced by Thompson in 1980. Since then, congruence numerical ranges and their radii have been studied by some authors. The purpose of this article is to discuss several aspects of this subject. We survey some known results and propose some problems for furthur study. A few new results are also presented.     

格的Fuzy同余理想

文［１］提出了格的Ｆｕｚｚｙ理想、Ｆｕｚｙ同余关系等概念并对其性质进行了探讨本文在此基础上提出了格的Ｆｕｚｚｙ同余理想的概念并分别讨论了格、Ｆｕｚｚｙ商格及格直积的Ｆｕｚｚｙ同余理想，得到了一些初步结果     

关于广义Stirling数偶的同余性

本文证明了广义Ｓｔｉｒｌｉｎｇ数偶的一些同余性质，从而回答了文［５］中的一个猜测．这些结果做为特例推广了已知的关于两类Ｓｔｉｒｌｉｎｇ数的同余性质．     

关于拟正则半环上可除半环同余的注记

研究了拟正则半环上的同余,给出了拟正则半环上一类理想的性质,并利用这类理想给出了拟正则半环上可除半环同余的几种等价形式.     

矩阵代数上的局部合同变换和局部相似变换

令F表示任意域,Mn（F）表示由F上所有n×n矩阵形成的结合代数.本文的目的是研究Mn（F）上具有如下性质的两类线性映射,其中一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个合同变换在该点的取值相同,另一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个相似变换在该点的取值相同,随着Mn（F）上的点不同,这些合同变换和相似变换可能也不同.利用矩阵的秩、幂等阵以及幂零阵的性质,通过矩阵计算的方法证明了第一类线性映射或者是合同变换或者是合同变换与转置变换的复合,第二类线性映射或者是相似变换或者是相似变换与转置变换的复合.由这个结果可知存在真正意义上的局部合同变换和局部相似变换,从而丰富了局部映射理论的研究。     

Note on a paper of thompson: the congruence numerical range

Given n×n Complex matrices A, Cdefine the C-congruence numerical range of A to be the set [ILM0001]. R.C. Thompson has characterized R_C(A) when [ILM0002] are fixed complex numbers. In this note. we obtain some analogous results about R_t(A) when C is skew symmmetric and a simple proof of the result of Thompson is given.Moreover, we characterize a certain set of partial off diagonals under congruence unitary transformation.     

幂格的同余关系

研究分配格的幂格的同余关系,给出幂格的同余关系格的一些重要性质,并得到幂格的一个同余类和同余关系格的两个交半格。     

对合Quantale同余关系

给出对合Quantale同余的概念,得到了对合Quantale同余的一些性质,讨论了对合Quantale同余与同态之间的关系,最后给出了对合Quantale乘积上的同余关系的结构。     

Remarks on annihilators preserving congruence relations

In this note we shall give some results on annihilators preserving congruence relations, or AP-congruences, in bounded distributive lattices. We shall give some new characterizations, and a topological interpretation of the notion of annihilator preserving congruences introduced in [JANOWITZ, M. F.: Annihilator preserving congruence relations of lattices, Algebra Universalis 5 (1975), 391–394]. As an application of these results, we shall prove that the quotient of a quasicomplemented lattice by means of a AP-congruence is a quasicomplemented lattice. Similarly, we will prove that the quotient of a normal latttice by means of a AP-congruence is also a normal lattice.     

Rectangular Ring Congruences on an E-inversive Semiring

In this paper, we discussed the property of rectangular band semiring congruence and ring congruence on a semiring and gave some characterizations and structure of rectangular ring congruence on an E-inversive semiring.     

群中模糊同余关系下的模糊粗糙群

在模糊同余关系下，首次提出了群中模糊同余关系下的模糊粗糙集和模糊粗糙群的概念，并讨论了它的一系列的性质，得到了一些有意义的结果。     

Results and problems on congruence lattice representations

We survey some results and problems on congruence lattice representations. We will consider congruence representations and similarity type, and we will consider simultaneous congruence representations. A seemingly innocuous problem unites the sections. This paper is dedicated to Walter Taylor. Received October 9, 2005; accepted in final form January 3, 2006.     

Good congruences on abundant semigroups with multiplicative quasi-adequate transversals

In this paper, some properties of good homomorphic images of abundant semigroups with a multiplicative quasi-adequate transversal are obtained. After that, good congruences on a class of abundant semigroups with such transversals are studied. We characterize the good congruences by using the so-called good congruence triple. Furthermore, we prove that any good congruence on a class of abundant semigroups with such transversals can be constructed by the good congruence triple.     

分配伪补格的主理想

本文给出了分配伪补格 ( L;∧ ,∨ ,* ,0 ,1 )中的主理想 I=( d]成为同余理想的充分必要条件 .当 L是局部有限时 (即 d∈ S( L) ,Fd={x|x* * =d}有限 ) ,对骨架 S( L)中的每个元素 d,我们找到了以 I=( d]为核心的最小同余关系 ,利用以上结果我们得到一个 Stone代数是布尔代数的一些等价条件 .     

关于左型A半群上的fuzzy同余

引入了左富足半群上fuzzy右好同余和fuzzy右消去同余的概念,给出了左富足半群上fuzzy右好同余的性质和特征.在此基础上,给出了左型A半群上fuzzy右好同余和fuzzy右消去同余的性质.得到了左型A半群上的fuzzy右好同余为fuzzy右消去同余的充要条件.     

R0-代数上的Fuzzy同余关系

在R0-代数中引入Fuzzy同余关系的概念，讨论了Fuzzy同余关系的一些基本性质，并通过对R0-代数中的Fuzzy MP滤子，特别是生成Fuzzy MP滤子的讨论建立了Fuzzy MP滤子与Fuzzy同余关系之间的联系。     

模糊半群上的模糊同余及其扩张

给出模糊半群上的模糊同余的概念，并进一步研究它的一些基本代数性质。同时研究带有模糊半群上的模糊同余扩张性质(FCEPF)的半群类，得到一个半群有模糊半群上的模糊同余扩张性质、有模糊同余扩张性质(FCEP)、有同余扩张性质(CEP)三个条件是等价的。     

One simple result concerning imprimitivity relations of monoids of transformations and its applications

A subsetX of an algebraA is called conditional if every congruence relation onA is uniquely determined by its congruence classes containing elements ofX. In terms of transformation monoids, it is found in which case one can be sure that a given subsetX is conditional. Some applications of this results to semigroups and lattices are demonstrated. These applications lead to known results as well as to new ones. Particularly, some sufficient conditions are obtained for semigroups and lattices to be harmonic.     

Graphical algebras — a new approach to congruence lattices

In 1970, H. Werner considered the question of which sublattices of partition lattices are congruence lattices for an algebra on the underlying set of the partition lattices. He showed that a complete sublattice of a partition lattice is a congruence lattice if and only if it is closed under a new operation called graphical composition. We study the properties of this new operation, viewed as an operation on an abstract lattice. We obtain some necessary properties, and we also obtain some sufficient conditions for an operation on an abstract lattice L to be this operation on a congruence lattice isomorphic to L. We use this result to give a new proof of Grätzer and Schmidt’s result that any algebraic lattice occurs as a congruence lattice.     

关于Clifford半群上的fuzzy同余

利用半群fuzzy同余的概念,讨论一类特殊的完全正则半群,即Clifford半群上的fuzzy同余.研究该类半群上fuzzy同余的性质.在此基础上,给出Clifford半群上fuzzy同余的性质和特征,得到Cllifford半群上fuzzy同余为fuzzy消去同余的充要条件.