具有Q-逆断面的正则半群的同余格

【作者】 商宇；

【导师】 汪立民；

【作者基本信息】 华南师范大学 ， 基础数学， 2005， 硕士

【摘要】 本文首先介绍了近二十年来具有Q-逆断面的正则半群的一些研究成果,接着针对具有Q-逆断面的正则半群S的关于基于子半群R和L为构件的结构,研究了具有Q-逆断面的正则半群上相应的同余及同余格.设S^o为正则半群S的逆子半群,若对任意x∈S有|V（x）∩S^o|=1,称S为具有逆断面S^o的正则半群,称S^o为S的逆断面,其中V（x）表x的所有逆元的集合。在这种情形下,V（x）∩S^o的唯一的元记为x^o且（x^o）^o记为x^oo。记S^o的幂等元的半格为E^o,记McAlister和McFadden在1982年给出了具有Q-逆断面S^o的正则半群S的结构定理,汪立民在1995年首先给出了S上相应的同余刻划,研究了同余格Con（S）上的五种同余关系T,T_T,T_l,U,V。1998年汪立民和唐西林把这种方法推广到了一般的具有逆断面S^o的正则半群的情形,从而彻底解决了具有逆断面的正则半群上的同余和同余格的刻划问题。Saito在1985年在文[11]中给出了具有Q-逆断面S^o的正则半群S的基于子半群R和L为结构分件的结构定理。 在第二章我们对具有Q-逆断面的正则半群S的基于子半群R,L为构件的结构,给出了R,L上的同余的相容条件,引入了用R和L上同余作成的同余对的概念,给出了S上的相应的同余刻划。用我们给出的同余刻划方法,描述了逆半群同余,群同余和幂等分离同余。我们给出了导出这些同余的格同态.我们定义了Con（S）上的等价关系W,Q,,它们都是Con（S）上的完全同余,这些完全同余的每一个类是区间,每一个类的极大,极小同余都用前面得到的刻划方法给出了表示。最后讨论了全关系ω的W,Q类的一些性质并给了Con（S）的一个子直积分解。更多还原

【Abstract】 In this paper ,we first give a survey of the research on the congruence of regular semigroups with inverse transversals .Then we study the congruence and congruence lattices of regular semigroups with Q-inverse transversals on the basis of the structure related the build bricks R and L.For a regular semigroup S, ainverse subsemigroup S° of S is an inverse transversal if |/V(x) ∩ S°| = 1 for each x ∈ S, where V(x) denotes the set of inverses of x. In this case ,the unique element of V{x) ∩ S° is denoted by x° and x~∞ denotes (x°)° E° denotes the semilattice of idempotents of S°. Some notations are defined as follows ,The structure of regular with Q- inverse transversals 5° was given by Mcalister and Mcfadden in 1982. Congruence lattices of regular semigroups with inverse transversals were first studied by Wang Limin. Wang Limin and Tang Xilin extended the method used above to investigate the congruence lattice to the general case .Saito gave the structure theorem of regular semigroups with Q-inverse transversals on the basis of the structure related the" build bricks" R and L .In the second chapter ,we study regular semigroups with Q-inverse transversals on the basis of the structure related the build bricks R and L. We will obtain the so called congruence pair based on Saito structure theorem, so that it produces a congruence on S abstractly. We consider the lattice of congruence pair ordered by componentwise inclusion and establish an isomorphism between the congruence lattice and the lattice of congruence pair .We also offer some special congruences on regular semigroups with Q-inversal transversal.Two complete congruence on the congruence lattice are analysed . The classes of each complete congruence which are intervals are discussed and their least and greast elements presented clearly. The extremal values of classes of complete congruence for the universal relation are exhibited and some of their relation are explored.Finally ,we arrive at a subdirect decompositions of the congruence lattice.更多还原