变换半群及图的自同态幺半群的研究 Alternative Title Some studies on transformation semigroups and endomorphism monoids of graphs 张佳 Thesis Advisor 罗彦锋 2017-09-30 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 变换半群 自同态 自然偏序 二部图 Cayley-图 正则半群 完全正则半群 富足半群 纯整半群 格林关系 星格林关系 自同态谱 自同态型 Abstract 本文主要研究了一类保序的变换半群及某些图的自同态幺半群的性质和结构, 全文共分为六章。第二章主要研究了一类带有限制集的保序变换半群 , 刻画了O(X,Y)的格林关系和星格林关系, 证明了 O(X,Y) 是富足半群但不是正则半群. 并刻画了O(X,Y) 的正则元, 给出了 O(X,Y)是正则半群的充分必要条件. 同时, 讨论了 O(X,Y)上的一些计数问题。第三章继续讨论带有限制集的保序变换半群O(X,Y), 研究其元素的自然偏序.讨论了O(X,Y)上元素的偏序关系, 确定了元素的相容性, 同时刻画了 O(X,Y)的极大元, 极小元以及两个元素的最大下界。第四章主要研究了二部图的广义字典积图的自同态幺半群是正则的充分必要条件. 给出了当时G=K1,n,是自同态正则的充分必要条件. 并在时d(G)>3, 分别定义了两个子图和两个K3子图的距离, 并分别刻画了在不包含子图K4时和包含K4图时,是自同态正则的充分必要条件。第五章主要研究一类4 正则的Cayley 图, 同时也是两个圈的伪笛卡尔积图. 刻画了两个圈的伪笛卡尔积图的自同构群, 以及自同态幺半群, 证明了其自同态幺半群同构当且仅当图同构, 即两个圈的伪笛卡尔积图可由其自同态幺半群所确定. 并描述了其自同态幺半群的代数结构, 同时给出了其自同态幺半群是正则半群, 完全正则半群和纯整半群的充分必要条件以及一些计数问题. 最后刻画了正则伪笛卡尔积图的自同态谱和自同态型。第六章主要研究了8-图的正则自同态幺半群的代数结构, 确定了正则自同态幺半群的格林关系, 正则-D类, 以及相关的一些计数问题. 同时也讨论了8-图的自同态幺半群的完全正则性. 最后给出了正则8-图的自同态谱和自同态型的刻画. Other Abstract This Ph.D. dissertation consists of six chapters. We study the properties and constructions of the certain order-preserving transformation semigroups and the endomorphism monoids of graphs. In chapter 2, a class of order-preserving transformation semigroups O(X,Y)   which preserve Y invariant is explored. We characterize the Green’s-relations and Green’s*-relations on , meanwhile show that O(X,Y)  is abundant but not a regular semigroups. Then we describe the regular elements of  and determine when  is a regular semigroup. Furthermore, we compute the cardinalities of E(O(X,Y)  ) ,Reg(O(X,Y)  ) and O(X,Y) , respectively. In chapter 3, we assgin O(X,Y) with so-called natural partial order. Under this partial order, we determine when two elements of O(X,Y)  are related, find the compatible  elements, the maximal elements, the minimal elements, and the greatest lower bound of two elements.In chapter 4, we determine the End-regular generalized lexicographic products of bipartite graphs G[Bx] . We show the conditions that when G[Bx] is End-regular for G=K1,n . For D(G)>3 , we give the sufficiency and necessity conditions for End-regular of G[Bx] via the definitions of the distance of two subgraphs K3 and two subgraphs K4, when G[Bx] does and does not contain of subgraphs K4. In chapter 5, we study the connected 4-regular Cayley graph by G by representing them as pseudo-cartesian product of two cycles. We characterize the automorphism groups and endomorphism monoids on  G, and prove that endomorphism monoids of pseudo-cartesian product of two cycles are isomorphism if and only if graphs are isomorphism.  We compute the cardinalities of  End(G ) and determine the condition under which  End(G ) is regular, completely regular and orthodox, respectively. Furthermore, we characterize the endomorphism spectrum and endomorphism type of End-regular. In chapter 6, we characterize the endomorphism monoids of 8-graphs. We show the Green’s relations and regular D-classes of endomorphism monoids of 8-graphs. Also we determine when endomorphism monoids of 8-graphs are completely regular semigroups, and compute the cardinalities of endomorphism monoids of 8-graphs. Furthermore, we characterize the endomorphism spectrum and endomorphism type of End-regular 8-graphs. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225609 Collection 数学与统计学院 Recommended CitationGB/T 7714 张佳. 变换半群及图的自同态幺半群的研究[D]. 兰州. 兰州大学,2017.
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