【作者】 李艳；

【导师】 刘妮；

【作者基本信息】 陕西师范大学 ， 基础数学， 2013， 硕士

【摘要】 Quantale是由C.J.Mulvey于1986年研究非交换的C*-algebra时引入的,其背景是给量子力学提供新的数学模型.在短短的二十几年中,Quantale理论受到了数学家和逻辑学家的密切关注,有关Quantale理论的大量新的观点及应用相继被给出.本文一方面对经典的Quantale理论展开继续研究,引入了伪连续余Quantale,同时对伪连续余Quantale的性质、理想、范畴做了细致而深入的研究.另一方面在模糊Quantale的基础上定义了模糊预Dual quantale和模糊预Girard quantale,研究了它们中的L-(预)核映射和L-(预)理想余核映射.同时也定义了模糊Dual quantale同态,研究了模糊Dual quantale范畴的性质.本文的主要内容安排如下：第一章预备知识.本章给出了将要用到的Quantale理论、余Quantale理论、范畴论以及模糊Quantale理论的相关概念和结论.第二章伪连续余Quantale首先,给出了伪连续余Quantale和伪半连续余Quantale的概念及其一些相关性质和结论.其次,引入了伪连续余Quantale中理想的概念,探论了伪连续余Quantale中理想的性质.最后,得到了伪连续余Quantale范畴的若干性质.第三章模糊预Dual quantale首先,给出了L-预核映射与L-预理想余核映射的概念及相关结论.其次,引入了模糊预Dual quantale和模糊预Girard quantale的概念,讨论了它们之间的关系.最后,探讨了模糊预Dual quantale上三-(预)核映射和L-(预)理想余核映射之间的关系,证明了模糊预Girard quantale上L-(预)核映射和L-(预)理想余核映射是一一对应的.第四章模糊Dual quantale范畴.首先,给出了模糊Dual quantale同态的概念,并且给出了模糊Dual quantale范畴等子的结构,证明了模糊Dual quantale范畴有乘积.其次,构造出了模糊Dual quantale范畴中的极限结构.最后,引入了模糊Dual 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】 The concept of quantale was introduced in1986by C.J.Mulvey to study noncommutative C*-algebra and provide a new mathematical model for quantum mechanics. In recent twenty years, the theory of quantale has been paid attention by mathematicians and logicists, lots of new ideas and applications of quantale have been established. The first part of this thesis is to further study classic quantale theory, the concept of pseudo continuous co-quantale is introduced, some properties, ideas and categorical properties of the pseudo continuous co-quantale are studied carefully and deeply. The second part of this thesis is to define the concepts of fuzzy pre-dual quantale and fuzzy pre-girard quantale on the base of fuzzy quantale and study the L-(pre-)nuclei and L-(pre-)ideal conuclei of them. At the same time, the concept of homomorphism of fuzzy dual quantale is introduced, some categorical properties of the fuzzy dual quantale are studied. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, we give some basic concepts and results of the quantale theory, co-quantale theory, category theory and the fuzzy quantale theory which will be used throughout the thesis.Chapter Two:Pseudo continuous co-quantale. Firstly, the concepts of pseudo continuous co-quantale and pseudo semi-continuous co-quantale are given, some properties of pseudo continuous co-quantale and pseudo semi-continuous co-quantale are studied. Secondly, the definition of ideal on the pseudo continuous co-quantale is introduced, the properties of ideal on the pseudo continuous co-quantale are stud-ied. Finally, some categorical properties of the pseudo continuous co-quantale are obtained.Chapter Three:Fuzzy pre-dual quantale. Firstly, the definitions of L-pre-nuclei and L-pre-ideal conuclei are given, some properties of them are studied. Secondly, the concepts of fuzzy pre-dual quantale and fuzzy pre-girard quantale are introduced, the relationship between fuzzy pre-dual quantale and fuzzy pre-girard quantale is discussed. Finally, the relationship between L-(pre-)nuclei and L-(pre-)ideal conuclei on the fuzzy pre-dual quantale are discussed, it is proved that there is a one-to-one correspondence between L-(pre-)nuclei and L-(pre-)ideal conuclei on the fuzzy pre-girard quantale.Chapter Four:The category of fuzzy dual quantale. Firstly, the concept of homomorphism of fuzzy dual quantale and the structure of equalizer of fuzzy dual quantale are given. Then, we prove that the category of fuzzy dual quantale has the product. Secondly, the limit on the category of fuzzy dual quantale is constructed. Finally, the concept of the inverse system on the category of fuzzy dual quantale is introduced, the inverse limit of the inverse system on the category of fuzzy dual quantale is constructed.更多还原