【作者】 李静；

【导师】 赵彬；

【作者基本信息】 陕西师范大学 ， 基础数学， 2006， 硕士

【摘要】 1986年，C．J．Mulvey在研究非交换的C~*-代数的谱时首先引入了Quantale的概念。从此，Quantale理论受到了数学家和逻辑学家的关注，1992年C．J．Mulvey和J．W．Pelletier在Quantale和C~*-代数理论的基础上提出了对合Quantale的概念，1993年S．Abramsky和S．Vickers提出了Quantale模的概念等等。Quantale自身具有丰富的序结构、代数结构和拓扑结构，与此相关的结构也有非常丰富的内容。本文研究了Quanale相关结构的性质，对Girard Quantale范畴的极限和逆极限作了较为细致而深入地研究。主要内容如下： 第一章 预备知识。本章给出了本文将要用到的Quantale理论、范畴理论的基本概念和结论。 第二章 Quantale相关结构的性质。本章首先研究了在幂等Quantale的条件下，代数Quantale与空间式Quantale的关系，得到了Quantale及其子Quantale是空间式的充分条件，并给出了子Quantale、商Quantale的若干例子。接着对Quantale矩阵进行了研究，讨论了幂等右侧Quantale上的幂零矩阵的若干性质，给出了幂等右侧Quantale上的矩阵为幂零矩阵的充要条件，得到了幂零矩阵的幂零指数的刻画定理。最后给出了Quantale模的余核映射的定义，得到了其与子Quantale模的对应关系；同时给出了Quantale上对偶双重模的定义，并研究了它的性质。 第三章 对合Quantale及其范畴中的定向极限。本章首先引入了关系对合Quantale的定义，得到了对合Quantale的表示定理，其次在范畴意义下，讨论了对合Quautale范畴与其满子范畴等价。最后，给出了对合Quantale范畴中定向极限的结构。 第四章 Girard Quantale范畴。本章研究了Girard Quantale范畴中的始对象、终对象等特殊对象，证明了此范畴不是点化范畴。给出了Girard Quantale范畴等化子的结构，证明了Girard Quantale范畴有乘积，并构造出了此范畴中的极限结构。最后给出了Girard Quantale范畴中逆系统的定义，得到了逆系统的逆极限结构。更多还原

【Abstract】 In 1986, C. J. Mulvey firstly introduced the concept of a quantale with the purpose of studying the spectrum of the noncommutative C~*—algebra. From then on, many mathematicians and logicists paid close attention to the theory of quantales. In 1992, C. J. Mulvey and J. W. Pelletier put forward a new concept--involutive quantale, basing on the theory of the quantale and C~*—algebra.And in 1993, S. Abramsky and S. Vickers proposed the concept of a quantale module. There are abundant contents in the structure of quantales, and so is the relative structure. The aim of this paper is to study some properties of the relative structure of quantale. At the same time, we also study carefully and deeply the limit and the inverse limit of the category of Girard quantales.The followings are the main contents of this paper:Chapter One Preliminary knowledge. In this chapter, we give the basic concepts and results of the theory of quantales and the category which are used in the whole paper.Chapter Two The properties of the relative structure of quantales. First of all, the relationship between algebraic quantales and spatial quantales is studied, under the circumstance of idempotent quantales. And the sufficient condition that quantales and their subquantales are spatial is obtained. Meenwhile, some examples of subquantales and quantic quotients are given. Next, quantale matrices are considered. We mainly discuss some properties of nilpotent matrices on idem-potent and right-sided quantales. At the same time, we obtain some important conclusion that the sufficient and necessary condition of nilpotent matrices on this quantales and the characteristic theorem of nilpotent index of nilpotent matrices. Finally, in the view of the definition of the conuclei of quantale modules, we find the correspondence between the conuclei of quantale modules and submodules over quantales. Then, the definition and properties of dual bimodule of quantale are given.Chapter Three Involutive quantale and the direct limit of the category of involutive quantales. Firstly, we introduce the definition of the relational involutive更多还原