Ω-范畴与模糊Domain中相关问题的研究
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摘要
Domain理论具有理论计算机科学与纯粹数学的双重研究背景,它是计算机程序设计语言指称语义学研究的数学基础,它与拓扑、逻辑、代数、范畴等学科有密切的联系.Domain理论将逼近与收敛的思想高度抽象化,其中序与拓扑的相互结合、相互作用是这一理论的基本特征.量化Domain在过去的三十年里经历了快速的发展,形成了Domain理论一个新的分支.它研究除了能提供定性信息还能提供定量信息的计算模型,比如能反映收敛的速度或程序的复杂度.目前量化Domain的研究已产生了众多不同的方法.其中,Ω-范畴作为研究量化Domain的一种方法受到了许多学者的关注.
Ω-范畴是包含偏序集与广义度量空间的一类特殊的enriched范畴.本文将对Ω-范畴相关结构及其在量化Domain理论中的应用展开研究.主要内容包含三个方面：一是将Ω-范畴与代数相结合,研究带有相容的Ω-范畴结构的代数结构；二是针对Ω-范畴研究中存在的问题,对Ω-范畴内在结构进行研究；三是研究Ω-范畴在量化Domain中的应用.具体内容安排如下：
第一章预备知识.本章介绍全文所需的预备知识,包括Domain理论中的基本概念、Ω-范畴中的有关概念与结论以及模糊Domain的概念.
第二章Ω-序代数结构.本章将Ω-范畴与代数结构相结合,考虑带有相容的Ω-范畴结构的代数结构.文中首先引入Ω-序半群的概念,给出大量的例子,并在其中引入同态与理想等基本概念.其次,基于Ω-伴随引入Ω-剩余序半群的概念,给出几个例子并讨论它的相关性质.最后,在Ω-序代数结构与带有模糊等于关系的代数结构构成的范畴之间建立了伴随关系.
第三章Ω-范畴中的几种基本结构.本章从三个方面对Ω-范畴研究中存在的问题与相关结构展开研究.第一节在L-完备格上引入L-完备格同余的概念,建立了它与L-闭包算子之间的关系；定义了L-完备格同余的商,并证明了一个L-完备格满同态的像同构于由该同态所诱导的同余的商.第二节是对Ω-范畴上反变Galois联络的研究,得到了Ω-范畴、Ω-范畴的张量积关于反变Galois联络的表示,并给出了反变Galois联络关于多值关系的表示.第三节是对基于Ω-范畴的多值拓扑的研究,本节在强L-拓扑范畴与L-frame范畴间建立了Stone型对偶.
第四章模糊Domain范畴的乘积.本章首先对L-偏序集中的几种完备性的关系进行研究,证明了一个L-偏序集是完备的当且仅当它是有限并完备且定向并完备的.在第二节讨论了模糊Domain与分明Domain的关系,给出了从模糊Domain诱导分明Domain,以及由分明Domain构造模糊Domain的方法与条件.第三节首先给出了模糊Domain的乘积中的模糊双小于关系的具体形式,进而证明了含最小元的cotensor完备的模糊Domain范畴和模糊连续格范畴有乘积.
第五章(代数)模糊连续格范畴的Cartesian闭性.本章针对Domain理论研究中的一个核心问题,即寻找模糊Domain范畴的Cartesian闭子范畴进行研究.主要证明了模糊连续格范畴与代数模糊连续格范畴是Cartesian闭的.本章首先回顾模糊Domain在几类模糊Scott连续的投射算子下的像的性质,主要证明了模糊Domain在模糊Scott连续的投射下的像仍是模糊Domain.进而研究模糊Domain的映射空间的连续性,基于第四章关于模糊连续格范畴乘积的结果,证明了模糊连续格的映射空间仍是模糊连续格,从而证明了模糊连续格范畴是Cartesian闭的.然后我们简要介绍了代数模糊Domain的有关概念与性质,讨论了代数模糊连续格的乘积与映射空间,并进一步证明了代数模糊连续格范畴也是Cartesian闭的.
With double backgrounds of theoretical computer science and pure mathematics, Domain Theory plays a fundamental role in the semantics of programming languages, and has close relations to topology, logic, algebra, category and some other mathematics disciplines. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way, and it is characterized by the close relation and interaction between orders and topologies. Quantitative Domain Theory (QDT for short) forms a new branch of Domain Theory and has undergone active research in the past three decades. QDT is concerned with models of computation that, in addition to qualitative information, allow also for the extraction of quantitative information—such as determining the speed of convergence or complexity of a program. At present several frameworks and approaches have been developed for QDT. Among them, the approach given by Ω-category theory attract a lot of attention.
Ω-categories are a special kind of enriched categories, and they include ordered sets and generalized metric spaces as specific examples. This dissertation concerns several structures related to Ω-categories, and the application of Ω-category theory in QDT. The content of this dissertation includes three aspects. Firstly, we study algebras with consistent Ω-category structures. Secondly, for several problems exist in Ω-category theory, we study the intrinsic structure of Ω-categories. Thirdly, we explore the application of Ω-category theory in QDT. The structure of this dissertation is organized as follows:
Chapter One:Preliminaries. This chapter gives some preliminaries that will be used throughout the thesis, including:basic concepts in domain theory, concepts and conclusions of Ω-category theory and the concept of fuzzy domain.
Chapter Two:Ω-ordered algebraic structures. In this chapter, we study algebras with consistent Ω-category structures. Firstly, the concept of Ω-ordered semigroup and some examples are given. Secondly, homomorphisms and ideals in Ω-ordered semigroups are studied. Thirdly, based on Ω-adjunction, the notion of Ω-residuated ordered semigroup and several examples about it are given, and some properties about it are studied. Lastly, the relation between Ω-ordered algebraic structures and algebras with fuzzy equalities is established.
Chapter Three:Basic structures about Ω-categories. Consisting of three sections, this chapter study several problems and structures about Ω-categories. In section one, we introduce the notion of many valued congruence relation in complete L-partially ordered sets. The relation between it and L-closure operator is established. We defined the quotient of congruence on complete L-partially ordered sets, and proved that the image of a surjective homomorphism of complete L-partially ordered sets is isomorphic to the quotient the congruence relation induced by that homomorphism. In Section two, we study contravariant Galois connections on Ω-categories. The representations of complete Ω-categories and tensor product of complete Ω-categories by contravariant Galois connections are given. And, it is proved that contravariant Galois connections on Ω-categories can be represented by certain Ω-valued relations. Section three is concerned with many valued topologies based on Ω-categories. A stone like dual is established between the category of strong L-topologies and the category of L-frames.
Chapter Four:Products of categories of fuzzy domains. In this chapter, first of all we study the relationships among three kinds of completeness of L-partially ordered sets. It is proved that an L-partially ordered set is complete if and only if it is both finite join complete and directed complete. Secondly, we study the relation between fuzzy domain and crisp domain. The methods and conditions of constructing them from each other are given. Thirdly, we give the concrete formation of fuzzy way below relation in product of fuzzy domains. Furthermore, we prove that the category of pointed cotensored fuzzy domains and the category of fuzzy continuous lattices have product.
Chapter Five:Cartesian closedness of the category of (algebraic) fuzzy continuous lattices. This chapter is devoted to search for Cartesian closed subcategories of fuzzy do-mains. It is mainly proved that the category of fuzzy continuous lattices and the category of algebraic fuzzy continuous lattices are Cartesian closed. In section one of this chapter, we recall some properties of the images of fuzzy domains under several fuzzy Scott con-tinuous projection operators. It is mainly proved that the image of a fuzzy domain under a fuzzy Scott continuous projection operator remains to be a fuzzy domain. Furthermore, we study the continuity of function spaces of fuzzy domains, based on the results about products of the category of fuzzy continuous lattices given in Chapter four, we prove that the function spaces of fuzzy continuous lattices are fuzzy continuous lattices. Therefore, we prove that the category of fuzzy continuous lattices is Cartesian closed. After intro-ducing the definition and some properties of algebraic fuzzy domains, we prove that the category of algebraic fuzzy continuous lattices have products and function spaces and the category of algebraic fuzzy continuous lattices is Cartesian closed.
Ω-范畴是包含偏序集与广义度量空间的一类特殊的enriched范畴.本文将对Ω-范畴相关结构及其在量化Domain理论中的应用展开研究.主要内容包含三个方面：一是将Ω-范畴与代数相结合,研究带有相容的Ω-范畴结构的代数结构；二是针对Ω-范畴研究中存在的问题,对Ω-范畴内在结构进行研究；三是研究Ω-范畴在量化Domain中的应用.具体内容安排如下：
第一章预备知识.本章介绍全文所需的预备知识,包括Domain理论中的基本概念、Ω-范畴中的有关概念与结论以及模糊Domain的概念.
第二章Ω-序代数结构.本章将Ω-范畴与代数结构相结合,考虑带有相容的Ω-范畴结构的代数结构.文中首先引入Ω-序半群的概念,给出大量的例子,并在其中引入同态与理想等基本概念.其次,基于Ω-伴随引入Ω-剩余序半群的概念,给出几个例子并讨论它的相关性质.最后,在Ω-序代数结构与带有模糊等于关系的代数结构构成的范畴之间建立了伴随关系.
第三章Ω-范畴中的几种基本结构.本章从三个方面对Ω-范畴研究中存在的问题与相关结构展开研究.第一节在L-完备格上引入L-完备格同余的概念,建立了它与L-闭包算子之间的关系；定义了L-完备格同余的商,并证明了一个L-完备格满同态的像同构于由该同态所诱导的同余的商.第二节是对Ω-范畴上反变Galois联络的研究,得到了Ω-范畴、Ω-范畴的张量积关于反变Galois联络的表示,并给出了反变Galois联络关于多值关系的表示.第三节是对基于Ω-范畴的多值拓扑的研究,本节在强L-拓扑范畴与L-frame范畴间建立了Stone型对偶.
第四章模糊Domain范畴的乘积.本章首先对L-偏序集中的几种完备性的关系进行研究,证明了一个L-偏序集是完备的当且仅当它是有限并完备且定向并完备的.在第二节讨论了模糊Domain与分明Domain的关系,给出了从模糊Domain诱导分明Domain,以及由分明Domain构造模糊Domain的方法与条件.第三节首先给出了模糊Domain的乘积中的模糊双小于关系的具体形式,进而证明了含最小元的cotensor完备的模糊Domain范畴和模糊连续格范畴有乘积.
第五章(代数)模糊连续格范畴的Cartesian闭性.本章针对Domain理论研究中的一个核心问题,即寻找模糊Domain范畴的Cartesian闭子范畴进行研究.主要证明了模糊连续格范畴与代数模糊连续格范畴是Cartesian闭的.本章首先回顾模糊Domain在几类模糊Scott连续的投射算子下的像的性质,主要证明了模糊Domain在模糊Scott连续的投射下的像仍是模糊Domain.进而研究模糊Domain的映射空间的连续性,基于第四章关于模糊连续格范畴乘积的结果,证明了模糊连续格的映射空间仍是模糊连续格,从而证明了模糊连续格范畴是Cartesian闭的.然后我们简要介绍了代数模糊Domain的有关概念与性质,讨论了代数模糊连续格的乘积与映射空间,并进一步证明了代数模糊连续格范畴也是Cartesian闭的.
With double backgrounds of theoretical computer science and pure mathematics, Domain Theory plays a fundamental role in the semantics of programming languages, and has close relations to topology, logic, algebra, category and some other mathematics disciplines. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way, and it is characterized by the close relation and interaction between orders and topologies. Quantitative Domain Theory (QDT for short) forms a new branch of Domain Theory and has undergone active research in the past three decades. QDT is concerned with models of computation that, in addition to qualitative information, allow also for the extraction of quantitative information—such as determining the speed of convergence or complexity of a program. At present several frameworks and approaches have been developed for QDT. Among them, the approach given by Ω-category theory attract a lot of attention.
Ω-categories are a special kind of enriched categories, and they include ordered sets and generalized metric spaces as specific examples. This dissertation concerns several structures related to Ω-categories, and the application of Ω-category theory in QDT. The content of this dissertation includes three aspects. Firstly, we study algebras with consistent Ω-category structures. Secondly, for several problems exist in Ω-category theory, we study the intrinsic structure of Ω-categories. Thirdly, we explore the application of Ω-category theory in QDT. The structure of this dissertation is organized as follows:
Chapter One:Preliminaries. This chapter gives some preliminaries that will be used throughout the thesis, including:basic concepts in domain theory, concepts and conclusions of Ω-category theory and the concept of fuzzy domain.
Chapter Two:Ω-ordered algebraic structures. In this chapter, we study algebras with consistent Ω-category structures. Firstly, the concept of Ω-ordered semigroup and some examples are given. Secondly, homomorphisms and ideals in Ω-ordered semigroups are studied. Thirdly, based on Ω-adjunction, the notion of Ω-residuated ordered semigroup and several examples about it are given, and some properties about it are studied. Lastly, the relation between Ω-ordered algebraic structures and algebras with fuzzy equalities is established.
Chapter Three:Basic structures about Ω-categories. Consisting of three sections, this chapter study several problems and structures about Ω-categories. In section one, we introduce the notion of many valued congruence relation in complete L-partially ordered sets. The relation between it and L-closure operator is established. We defined the quotient of congruence on complete L-partially ordered sets, and proved that the image of a surjective homomorphism of complete L-partially ordered sets is isomorphic to the quotient the congruence relation induced by that homomorphism. In Section two, we study contravariant Galois connections on Ω-categories. The representations of complete Ω-categories and tensor product of complete Ω-categories by contravariant Galois connections are given. And, it is proved that contravariant Galois connections on Ω-categories can be represented by certain Ω-valued relations. Section three is concerned with many valued topologies based on Ω-categories. A stone like dual is established between the category of strong L-topologies and the category of L-frames.
Chapter Four:Products of categories of fuzzy domains. In this chapter, first of all we study the relationships among three kinds of completeness of L-partially ordered sets. It is proved that an L-partially ordered set is complete if and only if it is both finite join complete and directed complete. Secondly, we study the relation between fuzzy domain and crisp domain. The methods and conditions of constructing them from each other are given. Thirdly, we give the concrete formation of fuzzy way below relation in product of fuzzy domains. Furthermore, we prove that the category of pointed cotensored fuzzy domains and the category of fuzzy continuous lattices have product.
Chapter Five:Cartesian closedness of the category of (algebraic) fuzzy continuous lattices. This chapter is devoted to search for Cartesian closed subcategories of fuzzy do-mains. It is mainly proved that the category of fuzzy continuous lattices and the category of algebraic fuzzy continuous lattices are Cartesian closed. In section one of this chapter, we recall some properties of the images of fuzzy domains under several fuzzy Scott con-tinuous projection operators. It is mainly proved that the image of a fuzzy domain under a fuzzy Scott continuous projection operator remains to be a fuzzy domain. Furthermore, we study the continuity of function spaces of fuzzy domains, based on the results about products of the category of fuzzy continuous lattices given in Chapter four, we prove that the function spaces of fuzzy continuous lattices are fuzzy continuous lattices. Therefore, we prove that the category of fuzzy continuous lattices is Cartesian closed. After intro-ducing the definition and some properties of algebraic fuzzy domains, we prove that the category of algebraic fuzzy continuous lattices have products and function spaces and the category of algebraic fuzzy continuous lattices is Cartesian closed.
引文
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