图的强化缩核与图的强自同态幺半群的正则性

本文研究图及其强自同态幺半群，首先刻画了图的强自同态幺半群的正则元，然后给出了此幺半群正则的充要条件，这推广了（１）和（２）中关于有限图的强自同态幺半群正则的结果。     

图的强收缩核与图的强自同态幺半群的正则性

本文研究图及其强自同态幺半群．首先刻画了图的强自同态幺半群的正则元，然后给出了此幺半群正则的充要条件．这推广了［１］和［２］中关于有限图的强自同态幺半群正则的结果．     

一族具有正则幺半群的图

本文给出了一族具有正则自同态幺半群的图及相应的自同态计数公式。     

图的完全正则自同态

作为图的代数分析的一部分，对图的自同态幺半群的研究近年来有一定的进展（参见［３］及［４］）。这类研究的主要目的在于将半群理论应用于图论。文献［５］研究了图的正则自同态及其逆。在此基础上本文进一步描述了图的完全正则自同态的组合特征；同时对含有完全正则自同态ｆ的极大子群，文中也明确给出了其单位元素及ｆ的逆     

广义字典序积的自同态幺半群

讨论了图的广义字典序积的自同态幺半群的性质,给出了广义字典序积图X[Yz|x∈V（X）]的自同态幺半群与X,Yx(x∈V(X))的自同态幺半群的圈积相等的充要条件。     

树的联的end-正则性

明确给出了具有正则自同态幺半群的两个树的联．     

具有完全正则自同态半群的分裂图

本文给出了具有完全正则自同态半群的分裂图的结构特征．其证明方法有望应用于其他图族自同态半群的正则性及完全正则性的研究．     

END-正则的循环图

称图X是End-正则图如果它的自同态幺半解EndX是正则的幺半解，即关于任意自同态f存在一个自同态g使得fgf=f。本文对顶点度数小于4的End-正则循环图进行了刻划。     

右消去幺半群、左正则带和左正则型A幺半群

本文利用右消去幺半群,左正则带建立了真左正则型A幺半群．在证明了任一左正则型A幺半群均有P－覆盖后,给出P－覆盖的结构．     

L—逆左系对幺半群的刻画

正则左Ｓ－系是ｖｏｎ ｎｅｕｍａｎｎ正则半群的自然推广，逆左Ｓ－系是逆半群的自然扩广，作为左逆半群的自然推广，本文引入了Ｌ－逆左系的概念，并用来刻画了几类幺半群，如左逆幺半群，逆幺半群，ａｄｅｑｕａｔｅ幺半群等。     

Cayley graphs as classifiers for data mining: The influence of asymmetries

The endomorphism monoids of graphs have been actively investigated. They are convenient tools expressing asymmetries of the graphs. One of the most important classes of graphs considered in this framework is that of Cayley graphs. Our paper proposes a new method of using Cayley graphs for classification of data. We give a survey of recent results devoted to the Cayley graphs also involving their endomorphism monoids.     

The Krull-Schmidt Theorem in the Case Two

We study what happens if, in the Krull-Schmidt Theorem, instead of considering modules whose endomorphism rings have one maximal ideal, we consider modules whose endomorphism rings have two maximal ideals. If a ring has exactly two maximal right ideals, then the two maximal right ideals are necessarily two-sided. We call such a ring of type 2. The behavior of direct sums of finitely many modules whose endomorphism rings have type 2 is completely described by a graph whose connected components are either complete graphs or complete bipartite graphs. The vertices of the graphs are ideals in a suitable full subcategory of Mod-R. The edges are isomorphism classes of modules. The complete bipartite graphs give rise to a behavior described by a Weak Krull-Schmidt Theorem. Such a behavior had been previously studied for the classes of uniserial modules, biuniform modules, cyclically presented modules over a local ring, kernels of morphisms between indecomposable injective modules, and couniformly presented modules. All these modules have endomorphism rings that are either local or of type 2. Here we present a general theory that includes all these cases.     

Endomorphism Spectra of Bipartite Graphs with Diameter Three and Girth Six

There are different endomorphisms for a graph. For a more systematic treatment of these endomorphisms the endomorphism spectrum and the endomorphism type of a graph are defined. Knauer characterized trees using their endomorphism types. The endomorphism type of bipartite graphs with diameter three and girth six is given in this paper.AMS Subject Classification (1991): 05C25 20M20Supported by the National Natural Science Foundation of China (19901012)     

Graphs whose endomorphism monoids are regular

In this paper, we give several approaches to construct new End-regular (-orthodox) graphs by means of the join and the lexicographic product of two graphs with certain conditions. In particular, the join of two connected bipartite graphs with a regular (orthodox) endomorphism monoid is explicitly described.     

Endomorphisms of graphs II. Various unretractive graphs

In this part of the article we investigate graphs for which different endomorphism monoids coincide. We consider endomorphisms, strong endomorphisms and automorphisms. Coincidences are investigated for joins of graphs and some lexicographic products. In an additional section the graphs with the respective properties are listed up to 8 vertices in two cases and up to 5 vertices in the remaining case.     

Semigroups of mappings on graphs

At present, there are quite a few investigations in the theory of semigroups devoted to semigroups of mappings on graphs. Up to now, endomorphism semigroups of graphs, extensive transformation semigroups of graphs, coloring semigroups of graphs and other semigroups of special mappings on graphs have been studied. The results obtained show the way graphs are determined by the above-mentioned semigroups. They also show the structure of semigroups of mappings and interrelations between properties of graphs and corresponding properties of semigroups associated with the graphs. This paper gives a survey of the main results in this field.     

一族具有正则绐幺半群的图

本文给出了一族具有正则自同态么半群的图及相应的自同态计数公式．     

图的字典序积和自同态幺半群

Ｆ．Ｈａｒａｒｙ￣［１］和Ｇ．Ｓａｂｉｄｕｓｓｉ￣［２］考虑过图Ｘ和ｙ的字典序积Ｘ［Ｙ］的自同构群ＡｕｔＸ［Ｙ］与它们各自的自同构群的圈积ＡｕｔＸ［ＡｕｔＹ］的关系，并给出了两者相等的一种刻划．在本文，我们考虑更广意义上的问题，即Ｘ［Ｙ］的自同态幺半群ＥｎｄＸ［Ｙ］与各自的自同态幺半群的圈积ＥｎｄＸ［ＥｎｄＹ］的关系，也给出了两者相等的一种刻划，同时得到了下面结果：如果Ｘ和Ｙ都是不含Ｋ＿３导出子图的连通图，且其中之一图有奇数围长，那么ＥｎｄＸ［Ｙ］＝ＥｎｄＸ［ＥｎｄＹ］．