伪补分配格的同余理想

发布时间：2018-09-03 10:05

【摘要】：格是序结构和代数结构的结合体.从布尔格在命题演算和开关理论中的重要作用可以看出格的重要.近年来由于有序理论在组合数学、Fuzzy数学中的广泛应用,使得格理论逐步发展成为现代数学的重要分支之一.伪补是格中补元的延伸,将格中补元满足的∧、∨两个运算减为一个运算∧就得到伪补.伪补的重要性在于:(1)有了伪补,原来不存在补元的元素却存在伪补元,这就扩大了格中元素存在补元的范围;(2)伪补的引入可提升格代数结构,即在代数结构上加上伪补可产生新的代数结构.将伪补融入格即成为伪补格,与一般的格有所不同,例如,一般格上的同余关系是对∧,∨具有替换性质,而伪补格上的同余关系则需要对∧,∨,*都具有替换性质,介于伪补的重要性,伪补格也成为人们热议的课题.理想在众多代数结构中都占据着主导地位,伪补格上的理想近年来是人们竞相追逐的研究课题.将伪补融入格理想,会衍生出新的理想,同余理想就是其中之一,同余理想是认识伪补格和同余关系的重要工具,例如,以同余理想为载体,人们搞清楚了伪补MS代数的内部结构,这为进一步研究伪补代数提供了理论支持.本文在前人研究的基础上,对伪补分配格上的同余理想做了进一步的研究,得出一些有意义的结论.文章介绍了伪补分配格上同余理想的性质以及理想成为同余理想的条件,关于特殊的同余理想:O-理想,给出了它的性质和具体表示形式,并用自己的方法或改进的方法予以证明.文章主要分为三部分:第一部分:预备知识.介绍了伪补分配格上同余理想的意义、研究现状及创新点;给出了研究伪补分配格上的同余理想所用到的概念、引理及结果,其中包括:格、格理想、格同余关系、伪补格、同余理想、O-理想等定义和相关结论.第二部分:同余理想的性质.根据*-同余关系的定义,说明了格同余关系成为*-同余关系的条件,对以同余理想为同余类的最小的*-同余关系,给出了具体表达形式;阐述了伪补分配格上的同余理想的性质,根据性质得出伪补格中元素所满足的若干格等式.第三部分:理想成为同余理想条件.介绍了伪补分配格上理想成为同余理想的若干充要条件和等价条件以及理想、素理想和主理想成为同余理想的条件,说明了主理想作为同余理想所具有的性质;给出了由同余理想与余核滤子相互寻找的方法;针对特殊的同余理想:O-理想,讨论了它的性质和理想成为O -理想的条件;并给出几种具体的O -理想.
[Abstract]:Lattice is a combination of ordered structure and algebraic structure. From the important role of Burg in proposition calculus and switch theory, we can see the importance of lattice. In recent years, due to the extensive application of order theory in combinatorial mathematics and fuzzy mathematics, lattice theory has gradually developed into one of the important branches of modern mathematics. Pseudo complement is an extension of complement in lattices. If two operations of a complement satisfying in a lattice are reduced to one operation, a pseudo complement can be obtained. The importance of pseudo complement is as follows: (1) with pseudo complement, the elements that do not have complement have pseudo complement, which expands the scope of complement in lattice. (2) the introduction of pseudo complement can enhance the algebraic structure of lattice. That is to say, adding pseudo complement to algebraic structure can produce new algebraic structure. It is different from the general lattice that the pseudo-complement is merged into the lattice, for example, the congruence relation on the general lattice has the property of substitution for the 位 and the Timov, while the congruence relation on the pseudo-complement lattice needs to have the property of substitution for all the congruences on the pseudo-complement lattice, and the congruence relation on the pseudo-complement lattice needs to have the property of substitution for all of them. Because of the importance of pseudo complement, pseudo complement lattice has become a hot topic. Ideals occupy a dominant position in many algebraic structures. In recent years, ideals on pseudo-complement lattices have been pursued by people. One of them is congruence ideal, which is an important tool for understanding pseudo-complement lattice and congruence relations. For example, congruence ideal is the carrier of congruence ideal. The internal structure of pseudo-complement MS algebras has been clarified, which provides theoretical support for further study of pseudo-complement algebras. On the basis of previous studies, this paper makes a further study of congruence ideals on pseudo-complementary distributive lattices, and draws some meaningful conclusions. In this paper, we introduce the properties of congruence ideals on pseudo-complementary distributive lattices and the conditions under which ideals become congruence ideals. And with their own method or improved method to prove. The article is divided into three parts: the first part: preparatory knowledge. This paper introduces the significance of congruence ideals on pseudo-complementary distributive lattices, the present research situation and innovation points, and gives the concepts, Lemma and results used to study congruence ideals on pseudo-complementary distributive lattices, including: lattice, lattice ideals, lattice congruence relations, pseudo-complement lattices. The definition and related conclusions of congruence ideals. Part two: the properties of congruence ideals. According to the definition of the lattice-congruence relation, this paper explains the condition that the lattice congruence relation becomes the lattice-congruence relation, and gives the concrete expression form of the minimal congruence relation in which the ideal of congruence is regarded as the class of congruence. The properties of congruence ideals on pseudo-complement distributive lattices are expounded. According to the properties, some lattice equations satisfied by elements in pseudo-complement lattices are obtained. The third part: ideal becomes congruence ideal condition. This paper introduces some necessary and sufficient conditions for an ideal on a pseudo-complementary distributive lattice to be a congruence ideal and some equivalent conditions, as well as the conditions for a prime ideal and a principal ideal to be a congruence ideal, and explains the properties of the principal ideal as a congruence ideal. This paper gives the method of searching by congruence ideal and conuclear filter, discusses its properties and the conditions under which the ideal becomes O-ideal for the special congruence ideal: O- ideal, and gives several concrete O-ideals.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153.1

【参考文献】