偽補(bǔ)分配格的同余理想
發(fā)布時(shí)間：2018-09-03 10:05
【摘要】：格是序結(jié)構(gòu)和代數(shù)結(jié)構(gòu)的結(jié)合體.從布爾格在命題演算和開關(guān)理論中的重要作用可以看出格的重要.近年來由于有序理論在組合數(shù)學(xué)、Fuzzy數(shù)學(xué)中的廣泛應(yīng)用,使得格理論逐步發(fā)展成為現(xiàn)代數(shù)學(xué)的重要分支之一.偽補(bǔ)是格中補(bǔ)元的延伸,將格中補(bǔ)元滿足的∧、∨兩個(gè)運(yùn)算減為一個(gè)運(yùn)算∧就得到偽補(bǔ).偽補(bǔ)的重要性在于:(1)有了偽補(bǔ),原來不存在補(bǔ)元的元素卻存在偽補(bǔ)元,這就擴(kuò)大了格中元素存在補(bǔ)元的范圍;(2)偽補(bǔ)的引入可提升格代數(shù)結(jié)構(gòu),即在代數(shù)結(jié)構(gòu)上加上偽補(bǔ)可產(chǎn)生新的代數(shù)結(jié)構(gòu).將偽補(bǔ)融入格即成為偽補(bǔ)格,與一般的格有所不同,例如,一般格上的同余關(guān)系是對(duì)∧,∨具有替換性質(zhì),而偽補(bǔ)格上的同余關(guān)系則需要對(duì)∧,∨,*都具有替換性質(zhì),介于偽補(bǔ)的重要性,偽補(bǔ)格也成為人們熱議的課題.理想在眾多代數(shù)結(jié)構(gòu)中都占據(jù)著主導(dǎo)地位,偽補(bǔ)格上的理想近年來是人們競(jìng)相追逐的研究課題.將偽補(bǔ)融入格理想,會(huì)衍生出新的理想,同余理想就是其中之一,同余理想是認(rèn)識(shí)偽補(bǔ)格和同余關(guān)系的重要工具,例如,以同余理想為載體,人們搞清楚了偽補(bǔ)MS代數(shù)的內(nèi)部結(jié)構(gòu),這為進(jìn)一步研究偽補(bǔ)代數(shù)提供了理論支持.本文在前人研究的基礎(chǔ)上,對(duì)偽補(bǔ)分配格上的同余理想做了進(jìn)一步的研究,得出一些有意義的結(jié)論.文章介紹了偽補(bǔ)分配格上同余理想的性質(zhì)以及理想成為同余理想的條件,關(guān)于特殊的同余理想:O-理想,給出了它的性質(zhì)和具體表示形式,并用自己的方法或改進(jìn)的方法予以證明.文章主要分為三部分:第一部分:預(yù)備知識(shí).介紹了偽補(bǔ)分配格上同余理想的意義、研究現(xiàn)狀及創(chuàng)新點(diǎn);給出了研究偽補(bǔ)分配格上的同余理想所用到的概念、引理及結(jié)果,其中包括:格、格理想、格同余關(guān)系、偽補(bǔ)格、同余理想、O-理想等定義和相關(guān)結(jié)論.第二部分:同余理想的性質(zhì).根據(jù)*-同余關(guān)系的定義,說明了格同余關(guān)系成為*-同余關(guān)系的條件,對(duì)以同余理想為同余類的最小的*-同余關(guān)系,給出了具體表達(dá)形式;闡述了偽補(bǔ)分配格上的同余理想的性質(zhì),根據(jù)性質(zhì)得出偽補(bǔ)格中元素所滿足的若干格等式.第三部分:理想成為同余理想條件.介紹了偽補(bǔ)分配格上理想成為同余理想的若干充要條件和等價(jià)條件以及理想、素理想和主理想成為同余理想的條件,說明了主理想作為同余理想所具有的性質(zhì);給出了由同余理想與余核濾子相互尋找的方法;針對(duì)特殊的同余理想:O-理想,討論了它的性質(zhì)和理想成為O -理想的條件;并給出幾種具體的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.
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O153.1
本文編號(hào)：2219657
[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.
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O153.1
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