模糊Domain与模糊Quantale中若干问题的研究
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摘要
摘要随着计算机科学的迅速发展,关于计算机科学的数学基础研究越来越受到人们的重视,已成为数学和计算机科学研究者共同关注的领域.产生于上个世纪70年代初的Domain理论和80年代的Quantale理论正是这样的两个重要交叉领域,它们各自独立发展,但从共同的数学基础来看,二者均基于数学中三大基本结构之一的序结构理论,同时与拓扑、代数、范畴、逻辑等学科有着密切的联系.尽管Domain理论与Quantale理论有着各自不同的研究对象和特点,但它们在一些方面是相互渗透和相互影响的,例如Quantale理论在量化Domain理论中的应用.自2000年以来,模糊集理论被应用到量化Domain理论中,形成了模糊Domain理论.本文一方面是对模糊Domain理论展开进一步的研究,另一方面是将模糊集理论应用到Quantale理论中,进行Quantale理论的模糊化研究.本文的主要内容安排如下：
第一章预备知识.本章给出了与本文相关的格论、逻辑代数、范畴论以及模糊偏序集方面的概念和结论.
第二章模糊偏序集的并完备化.首先给出了模糊偏序集的并完备化的概念,证明了在等价的意义下,模糊偏序集(X,e)的并完备化是由L~X上的相容的模糊闭包算子完全决定的.其次研究了并完备化的万有性质.最后给出了模糊偏序集的Dedekind-MacNeille完备化的范畴刻画.
第三章Φ-连续的模糊偏序集.首先讨论了weight类的相关性质,给出了模糊完备格之间保模糊并映射的一些等价刻画.其次得到了Φ-连续的模糊偏序集的相关性质,讨论了Φ-连续的模糊偏序集在一些特殊映射下的像仍是Φ-连续的模糊偏序集.最后研究了L-滤子的模糊SΦ-收敛.
第四章Φ-代数的模糊偏序集.首先引入了Φ-代数的模糊偏序集的概念,得到了Φ-代数的模糊偏序集的一些性质.其次讨论了Φ-同态的性质,给出了模糊偏序集的分类定理,将Hoffmann在分明情形下的分类定理推广到饱和的weight类的框架下.最后研究了Φ-完备的模糊偏序集上的基和权,讨论了模糊偏序集范畴与Φ-代数的模糊偏序集范畴之间的关系,证明了以Φ-代数的模糊偏序集为对象,以Φ-同态为态射的范畴ΦAFPOSH等价于以模糊偏序集为对象,以保模糊序映射为态射的范畴FPOS还证明了以模糊偏序集为对象,以Φ-映射为态射的范畴FPOID对偶等价于以Φ-代数的模糊偏序集为对象,以Φ-态射为态射的范畴ΦAFPOSM·
第五章模糊Quantale首先通过模糊Galois伴随给出了模糊Quantale的定义,并给出了模糊Quantale的相关例子.研究了模糊Quantale上的核映射和余核映射.其次引入了模糊Girard quantale的概念,证明了模糊Girard quantale上的L-核映射和L-理想余核是一一对应的.最后讨论了模糊序半群的模糊Quantale完备化,证明了在同构的意义下,模糊序半群(S,·,e)的模糊Quantale完备化是由Ls上的拓扑模糊闭包算子完全决定的.
第六章模糊Quantale范畴.本章首先证明了模糊Frame范畴是模糊Quantale范畴的反射满子范畴.其次证明了模糊Quantale范畴同构于L-代数范畴.最后给出模糊Quantale范畴的极限和逆极限结构.
Abstract With the rapid development of computer science, people are paying more and more attention to the research about its mathematical foundations which have been the area of common intersets of mathematicians and computer scientists. Domain theory and quantale theory established in the early1970's and1980's, re-spectively, are just two very important cross fields. They developed independently, but they are based on order theory from the viewpoint of the common mathemati-cal foundation. Meanwhile, they make the close relationship with topology, algebra, category, logic and so on. Although domain theory and quantale theory have differ-ent research objects and characteristics, they are mutual penetration and influence in some aspects, such as the applications of quantale theory to quantitative domain theory. Since2000, fuzzy set theory have been applied to the quantitative domain theory, forming fuzzy domain theory. The first part of this thesis is to further inves-tigate fuzzy domain theory. The second part is to study the fuzzification of quantale theory. The structure of this thesis is organized as follows:
Chapter One: Preliminaries. In this chapter, some concepts and results will be used in this paper are given.
Chapter Two:Join-completions of fuzzy posets. Firstly, the definition of join-completions of fuzzy posets is given. It is proved that the join-completions of a fuzzy poset (X. e) up to equivalence are completely determined by consistent fuzzy closure operators on Lx. Secondly, the universal property of join-completions is studied. At last, a categorical characterization of the Dedekind-MacNeille completion for a fuzzy poset is presented.
Chapter Three:Φ-continuous fuzzy posets. Firstly, some related properties of weight classes are discussed. Some equivalent characterizations of preserving fuzzy joins mappings between fuzzy complete lattices are given. Secondly, some related properties of Φ-continuous fuzzy posets are obtained. We discuss some particular mappings under which the images of Φ-continuous fuzzy posets are still Φ-continuous fuzzy posets. At last, fuzzy SΦ-convergence of L-filters is studied.
Chapter Four:Φ-algebraic fuzzy posets. Firstly, the concept of Φ-algebraic fuzzy posets is introduced. Some properties of Φ-algebraic fuzzy posets are ob-tained. Secondly, we discuss some properties of Φ-homomorphisms and give a classification theorem of fuzzy posets, which generalizes Hoffmann's classification theorem to the framework of saturated class of weights. At last, bases and weights on Φ-complete fuzzy posets are studied. The relations between the category of Φ-algebraic fuzzy posets and the category of fuzzy posets are discussed. It is proved that ΦAFPOSH is equivalent to FPOS. It is also proved that FPOID is dual equivalent to ΦAFPOSM·
Chapter Five:Fuzzy quantales. Firstly, we introduce the concept of fuzzy quantales by means of fuzzy Galois connections and give some examples of fuzzy quantales. The quantic nucleus and quantic conucleus on fuzzy quantales are stud-ied. Secondly, the concept of Girard quantales is introduced. It is proved that L-quantic nucleus and L-ideal conucleus are one-to-one corresponding. Finally, fuzzy quantale completions of fuzzy ordered semigroups are discussed. It is shown that up to isomorphism, the fuzzy quantale completions of a fuzzy ordered semigroup (S1,·, e) are completely determined by topological fuzzy closure operators on LS.
Chapter Six:The category of fuzzy quantales. In this chapter, we prove that the category of fuzzy frames introduced by Yao is a full reflective subcategory of the category of fuzzy quantales. Secondly, it is shown that the category of fuzzy quantales is isomorphic to the category of L-algebras. Finally, we give the structures of the limit and the inverse limit of the category of fuzzy quantales.
第一章预备知识.本章给出了与本文相关的格论、逻辑代数、范畴论以及模糊偏序集方面的概念和结论.
第二章模糊偏序集的并完备化.首先给出了模糊偏序集的并完备化的概念,证明了在等价的意义下,模糊偏序集(X,e)的并完备化是由L~X上的相容的模糊闭包算子完全决定的.其次研究了并完备化的万有性质.最后给出了模糊偏序集的Dedekind-MacNeille完备化的范畴刻画.
第三章Φ-连续的模糊偏序集.首先讨论了weight类的相关性质,给出了模糊完备格之间保模糊并映射的一些等价刻画.其次得到了Φ-连续的模糊偏序集的相关性质,讨论了Φ-连续的模糊偏序集在一些特殊映射下的像仍是Φ-连续的模糊偏序集.最后研究了L-滤子的模糊SΦ-收敛.
第四章Φ-代数的模糊偏序集.首先引入了Φ-代数的模糊偏序集的概念,得到了Φ-代数的模糊偏序集的一些性质.其次讨论了Φ-同态的性质,给出了模糊偏序集的分类定理,将Hoffmann在分明情形下的分类定理推广到饱和的weight类的框架下.最后研究了Φ-完备的模糊偏序集上的基和权,讨论了模糊偏序集范畴与Φ-代数的模糊偏序集范畴之间的关系,证明了以Φ-代数的模糊偏序集为对象,以Φ-同态为态射的范畴ΦAFPOSH等价于以模糊偏序集为对象,以保模糊序映射为态射的范畴FPOS还证明了以模糊偏序集为对象,以Φ-映射为态射的范畴FPOID对偶等价于以Φ-代数的模糊偏序集为对象,以Φ-态射为态射的范畴ΦAFPOSM·
第五章模糊Quantale首先通过模糊Galois伴随给出了模糊Quantale的定义,并给出了模糊Quantale的相关例子.研究了模糊Quantale上的核映射和余核映射.其次引入了模糊Girard quantale的概念,证明了模糊Girard quantale上的L-核映射和L-理想余核是一一对应的.最后讨论了模糊序半群的模糊Quantale完备化,证明了在同构的意义下,模糊序半群(S,·,e)的模糊Quantale完备化是由Ls上的拓扑模糊闭包算子完全决定的.
第六章模糊Quantale范畴.本章首先证明了模糊Frame范畴是模糊Quantale范畴的反射满子范畴.其次证明了模糊Quantale范畴同构于L-代数范畴.最后给出模糊Quantale范畴的极限和逆极限结构.
Abstract With the rapid development of computer science, people are paying more and more attention to the research about its mathematical foundations which have been the area of common intersets of mathematicians and computer scientists. Domain theory and quantale theory established in the early1970's and1980's, re-spectively, are just two very important cross fields. They developed independently, but they are based on order theory from the viewpoint of the common mathemati-cal foundation. Meanwhile, they make the close relationship with topology, algebra, category, logic and so on. Although domain theory and quantale theory have differ-ent research objects and characteristics, they are mutual penetration and influence in some aspects, such as the applications of quantale theory to quantitative domain theory. Since2000, fuzzy set theory have been applied to the quantitative domain theory, forming fuzzy domain theory. The first part of this thesis is to further inves-tigate fuzzy domain theory. The second part is to study the fuzzification of quantale theory. The structure of this thesis is organized as follows:
Chapter One: Preliminaries. In this chapter, some concepts and results will be used in this paper are given.
Chapter Two:Join-completions of fuzzy posets. Firstly, the definition of join-completions of fuzzy posets is given. It is proved that the join-completions of a fuzzy poset (X. e) up to equivalence are completely determined by consistent fuzzy closure operators on Lx. Secondly, the universal property of join-completions is studied. At last, a categorical characterization of the Dedekind-MacNeille completion for a fuzzy poset is presented.
Chapter Three:Φ-continuous fuzzy posets. Firstly, some related properties of weight classes are discussed. Some equivalent characterizations of preserving fuzzy joins mappings between fuzzy complete lattices are given. Secondly, some related properties of Φ-continuous fuzzy posets are obtained. We discuss some particular mappings under which the images of Φ-continuous fuzzy posets are still Φ-continuous fuzzy posets. At last, fuzzy SΦ-convergence of L-filters is studied.
Chapter Four:Φ-algebraic fuzzy posets. Firstly, the concept of Φ-algebraic fuzzy posets is introduced. Some properties of Φ-algebraic fuzzy posets are ob-tained. Secondly, we discuss some properties of Φ-homomorphisms and give a classification theorem of fuzzy posets, which generalizes Hoffmann's classification theorem to the framework of saturated class of weights. At last, bases and weights on Φ-complete fuzzy posets are studied. The relations between the category of Φ-algebraic fuzzy posets and the category of fuzzy posets are discussed. It is proved that ΦAFPOSH is equivalent to FPOS. It is also proved that FPOID is dual equivalent to ΦAFPOSM·
Chapter Five:Fuzzy quantales. Firstly, we introduce the concept of fuzzy quantales by means of fuzzy Galois connections and give some examples of fuzzy quantales. The quantic nucleus and quantic conucleus on fuzzy quantales are stud-ied. Secondly, the concept of Girard quantales is introduced. It is proved that L-quantic nucleus and L-ideal conucleus are one-to-one corresponding. Finally, fuzzy quantale completions of fuzzy ordered semigroups are discussed. It is shown that up to isomorphism, the fuzzy quantale completions of a fuzzy ordered semigroup (S1,·, e) are completely determined by topological fuzzy closure operators on LS.
Chapter Six:The category of fuzzy quantales. In this chapter, we prove that the category of fuzzy frames introduced by Yao is a full reflective subcategory of the category of fuzzy quantales. Secondly, it is shown that the category of fuzzy quantales is isomorphic to the category of L-algebras. Finally, we give the structures of the limit and the inverse limit of the category of fuzzy quantales.
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