半连续dcpo与伪相容连续Domain的若干研究

发布时间：2018-05-28 16:22

本文选题：连续格 + 半连续格　；参考：《淮北师范大学》2015年硕士论文

【摘要】：自从19世纪七十年代连续格与代数格被发现以后,连续格理论便成为了极为活跃的研究领域,对连续格理论的拓展也逐渐加深.半素理想是Y.Ray最先提出的并得到了进一步丰富；伍秀华等通过利用半素理想,定义出来另外的一种新型关系,并运用到半连续格当中；同样可以进行等价刻画的还有半连续dcpo或相容连续Domain的性质,而在本文第三章节把这些思想延伸到愈发宽泛的相关理论的拓展当中.自从D.S.Scott首次提出了Domain的概念,国内外数学家对Domain理论便产生极大的兴趣并进行深入研究,得到了连续Domain的许多类似于连续格的性质.第四、五章从广义的角度,从连续格的内蕴特征出发引入了伪相容连续Domain、伪相容连续Domain局部基的相关概念及性质.本文主要包括六个部分：第一章：绪论知识.主要介绍了本论文的研究现状、研究背景、研究意义,并扼要概述了本文所要研究的主要内容.第二章：预备知识.将全文将要用到的连续Domain理论相关的概念与结果等预备知识作了简要概述.第三章：本章研究半连续dcpo的局部半基的若干理论.主要研究连续格,半连续格.半连续dcpo强连续dcpo等的相关定义与主要性质的内部联系以及拓展方法,通过弱化与强化,比较拓展,得到它们的联系与区别,并做出一些补充.而对于连续偏序集,相容连续偏序集,弱Domain, exact Domain等的一些结论也类似可得.最后给出半连续dcpo的局部半基的概念,从而得到一些相关性质与结论.第四章：本章研究伪相容连续Domain本章根据相容Domain,引出了伪相容连续Domain与它的基的概念.进一步得出若干等价命题及相关性质的证明,从而将得出许多良好的性质与结论,从而对Domain理论作出进一步推广.而在本章最后介绍了在伪相容连续偏序集中一类特殊的相容定向集.第五章：本章研究伪相容连续Domain局部基的若干性质.本章在上一章的基础上引入相容定向集的容元,容集,以及伪相容连续Domain的局部基的概念,在此基础上讨论了伪相容连续Domain中容元的局部基的特点,并对伪相容连续Domain的局部基的其它重要相关性质进一步探讨.第六章：作为格理论中的一类比较特殊的元素,交既约元具有一些很好的性质,在格论中也占有重要的地位.本章通过引入连续交既约元的概念,对交既约元的性质展开探讨.
[Abstract]:Since the discovery of continuous lattices and algebraic lattices in the 1870s, the theory of continuous lattices has become a very active research field, and the extension of the theory of continuous lattices has gradually deepened. The semi-prime ideal was first put forward by Y.Ray and further enriched, Wu Xiuhua and others defined another new relationship by using semi-prime ideal, and applied it to semicontinuous lattice. There are also the properties of semicontinuous dcpo or compatible continuous Domain which can also be characterized as equivalent. In the third chapter of this paper we extend these ideas to the extension of more and more extensive related theories. Since D.S.Scott first put forward the concept of Domain, mathematicians at home and abroad have had great interest in the theory of Domain and studied it in depth. Many properties of continuous Domain similar to continuous lattices have been obtained. In the fourth and fifth chapters, the concepts and properties of pseudo-compatible continuous Domain local bases are introduced from the generalized point of view and from the intrinsic characteristics of continuous lattices. This article mainly includes six parts: chapter one: introduction knowledge. This paper mainly introduces the research status, research background, research significance, and briefly summarizes the main contents of this paper. Chapter two: preparatory knowledge. The concepts and results of continuous Domain theory which will be used in this paper are briefly summarized. Chapter 3: in this chapter, we study some theories of local semilobars of semicontinuous dcpo. We mainly study continuous lattice and semi-continuous lattice. The relative definitions of semicontinuous dcpo strongly continuous dcpo and the internal relations of its main properties, as well as its extension methods, are obtained by weakening and strengthening, comparing and expanding, and some additions are made. For continuous partial ordered sets, consistent continuous partial ordered sets, weak Domain, exact Domain and so on, some results are similar. Finally, the concept of local semi-basis of semicontinuous dcpo is given, and some related properties and conclusions are obtained. Chapter 4: this chapter studies pseudo-compatible continuous Domain. According to compatible domain, we introduce the concept of pseudo-compatible continuous Domain and its basis. Some equivalent propositions and related properties are further proved, and many good properties and conclusions will be obtained, thus further generalizing the Domain theory. At the end of this chapter, we introduce a class of special compatible orientations in pseudo-compatible continuous partial ordering sets. Chapter 5: in this chapter, we study some properties of pseudo-compatible continuous Domain local basis. On the basis of the previous chapter, this chapter introduces the concepts of the local basis of the compatible direction-set, the capacitive set, and the pseudo-compatible continuous Domain. On this basis, the characteristics of the local basis of the component in the pseudo-compatible continuous Domain are discussed. The other important properties of the local basis of pseudo-compatible continuous Domain are further discussed. Chapter 6: as a kind of special element in lattice theory, the intersection irreducible element has some good properties and plays an important role in lattice theory. By introducing the concept of continuous intersection irreducible elements, this chapter discusses the properties of intersection irreducible elements.
【学位授予单位】：淮北师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O153.1

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