【作者】 韩胜伟；

【导师】 赵彬；

【作者基本信息】 陕西师范大学 ， 基础数学， 2005， 硕士

【摘要】 Quantale概念是由C. J. Mulvey于1986年在研究非交换的C~*-algebra的谱时引入的,其背景是给量子力学提供新的模型。对它的研究涉及到非交换的C~*-algebra,环的理想理论,逻辑和计算机科学等诸多领域。Quantale可以看作是Frame的一般化,加之其丰富的序结构,代数结构和拓扑结构,因此其自身的内在结构也有极其丰富的内容。本文对Quantale中的核映射,Quantale商与Quantale同余三者之间的关系,以及单纯Quantale的结构,平凡Quantale的结构,真子Quantale, Quantale中的理想和Quantale矩阵等做了研究。主要内容如下: 第一章 预备知识。本章给出了有关Quantale的基本概念和结论。 第二章 单纯Quantale,本章研究了核映射,Quantale商和Quantale同余三者之间的关系;得到了一类Quantale是单纯Quantale的充要条件;给出了单纯Quantale的商式刻划,以及单纯Quantale的另一种描述。 第三章 Quantale中的理想。本章第一部分对子Quantale及其相关性质进行了讨论。首先我们给出了构造子Quantale的两种方法,其次研究了子Quantale的性质,同时我们利用子Quantale对Quantale中的素元进行了讨论。第二部分我们从映射角度给出了平凡Quantale的概念,并且给出了平凡Quantale的具体刻划;同时对有限Quantale的子结构进行了研究,并且我们进一步讨论了真子Quantale的存在性及其相关性质。第三部分对Quantale中的理想进行了讨论,给出了由Quantale中任意子集合生成(左,右)理想的具体结构;同时我们进一步研究了(左,右)理想与映射之间的关系,从而给出了(左,右)理想余核的概念,得到一个映射是理想余核的充要条件。 第四章 Quantale上的矩阵。本章首先给出了Quantale矩阵及其行列式的概念,并且研究了Quantale矩阵及其行列式的相关性质。其次,我们给出了Quantale矩阵的逆与广义逆的定义,同时对Quantale矩阵的逆与广义逆进行了较为系统的研究,得到了Quantale矩阵可逆的等价刻划。更多还原

【Abstract】 The concept of quantale was introduced by C. J. Mulvey in 1986 with the purpose of studying the spectrum of C*-algebra, as well as constructive foundations for quantum mechanics. The research of these subjects has related to several research area such as non-commutative C*-algebra, the ideal theory of rings, linear logic and theoretic computer science. There are abundant contents in the structure of quantales, because quantale can be regard as the generalization of frame. This paper analyses and studied the relations among nucleus, quantic quotient and quantale congruence, and the structures of simple quantale and trivial quantale, non-trivial quantale, the ideals of quantale. The main content is as follows.Chapter One Preliminary knowledge. In this chapter the author gives the main concepts and results of quantale theory used in the whole article.Chapter Two Simple quantale and quantic quotient. In this chapter the author discusses the relations among nucleus, quantic quotient and quantale congruence, obtains the equivalent condition that a kind of quantale is a simple quantale, gives the quotient characterization of simple quantale and a new definition of simple quantale.Chapter Three The ideals of quantale. In first part of this chapter, the author studies subquantale and its properties. Firstly, we give two ways to construct subquantales, study the properties of subquantales , and in the meantime we discuss the prime element of quantale by subquantale. The second part, we give the definition of the trivial quantale by the map , and give its concrete characterization; We study the subquantale of finite quantale, and prove the existence of non-trivial subquantale; In three part we study the ideal of quantale, and give the concrete construction of the ideal generated by an arbitary subset. Furthermore we study the relation between the ideal and the map, and obtain the equivalent condition that a map is an ideal conucleus.Chapter Four Quantale Matrix. In this chapter, we give the concepts of quantale matrix and the deterninant of quantale matrix , and study the properties of quantale matrix and the deterninant of quantale matrix. In the meantime, we also give the defintions of the inverse and the generalized inverse of quantale matrix,更多还原