非标准分析方法在模糊拓扑学中的若干应用
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摘要
非标准分析是使用非标准模型研究各种数学问题的新的数学理论.自A.Robinson于1961年创立非标准分析理论之后,人们把实数域及其上的各种关系称为分析的标准模型.在分析的标准模型中,或者说在实数域上展开的分析学称为标准分析.把实数域及其上关系的扩大称为分析的非标准模型.在分析的非标准模型中,实数域R的真扩张称为超实数域,记为*R.在非标准模型中,或者说在超实数域*R上展开的分析学称为非标准分析.非标准分析是标准分析的真扩张,非标准分析与标准分析的不同之处在于超实数域*R内包含无穷小的非零数及无穷大的数.
本论文主要研究了非标准模型的理论及其在模糊拓扑学中的若干应用,论文研究的主要目的有两个,一是尝试着将非标准模型归结为超结构,便于二次模型或多次模型的构造,二是尝试着将非标准分析方法应用于模糊拓扑学中,为非标准分析理论和模糊拓扑学理论的结合提供可能.这些尝试不仅完善了非标准分析理论,丰富了非标准分析方法的应用和研究领域,而且为模糊拓扑学的研究提供了一种新思路和新方法.论文具体研究内容如下：
(1)定义了一个集类上的个体集S及其上超结构V(S),证明了超结构V(S)是一个足够大的集合.以数理逻辑的一阶形式语言为基础,借助模型论中的解释映射提出了抽象非标准模型、具体非标准模型以及准非标准模型的概念.以超滤子为工具,以超幂的构造方法,构造了一个新的超结构V(*S),利用超滤子的特性,证明了新的超结构V(*S)是V(S)的一个非标准模型,指明了超滤子及有界量词句子在转换原理中的必要性.从几个角度说明了非标准分析理论中的*映射的构造,及其保持Boolean运算的性质.这样从一个超结构出发,将非标准模型又归结为了超结构,为进一步讨论二次模型打下了基础.
(2)给出超结构V(*S)中实体的概念,指出标准实体、内实体和外实体的差别,说明了常见记号*V(sS的符号性.定义了k-扩大模型,从k-充足的滤子出发,证明了k-扩大模型的存在性,并以此得到的超结构V(*S)是k-扩大模型的充分且必要条件,以及k-扩大模型的一些有趣的性质.相仿Luxemburg的方法,得到了k-饱和模型的存在性,讨论了k-饱和模型的充分且必要条件,以及k-饱和模型的性质.在这些基础上,证明了二次非标准模型V(**S)的存在性,定义了V(**S)中的一些实体,如二次标准实体、二次内实体等,利用这些实体,讨论了二次非标准模型的扩大性和饱和性.为将来进一步讨论多次非标准模型提供了基本的研究思路和方法.
(3)讨论了完备格L与其非标准扩张*L的关系,定义了内完备格,k-拷贝完备格与k-完备格,特别是k-完备格,对于一个格中的任意非空子集A,若|A|扩大模型和饱和模型的模糊表现形式,以及一些性质.这些准备为利用非标准分析方法研究模糊拓扑学提供了可能.
(4)定义了非标准模糊集合的概念,讨论了全体非标准模糊集合之族*[0,1]*X及其一些子族,如标准模糊集合之族σ[0,1]X)、内模糊集合之族*([0,1]X)等的性质.利用标准部分映射st将非标准模糊集合与模糊集合之间建立了关系.以模糊拓扑空间(X,δ)为基础,讨论了非标准模糊拓扑空间(*X,δ)的超紧性,并给出了模糊拓扑空间(X,δ)一种自然的Stone-Cech超紧化(X,ε)的构造方式.
(5)以区间数集Ⅱ(R)及其上的序关系为基础,定义了超区间数集Ⅱ(*R),并在其子集Ns(Ⅱ(R))上定义了一个等价关系～,证明了Ⅱ(R)与Ns(Ⅱ(*R))/～是序同构的.讨论了区间值度量空间(X,ρ)的非标准扩张(*X,*ρ),证明了(*X,*ρ)是一个超区间值度量空间.研究了(*X,*ρ)的有限点集Fin(*X)和拟近标准点集Qns(*X)两类特殊点集,并以此给出了区间值度量空间(X,ρ)的完备化的一种构造方式.
(6)定义了X×X上的模糊滤子F及其单子m(F),给出了X×X上的一个模糊滤子是X上的模糊一致结构的充分且必要条件,即m(F)是*X上的一个模糊等价关系.讨论了模糊一致空间(X,u)的非标准扩张(*X,u)的性质,证明了(*X,u)是一个模糊一致空间,并且它是(X,u)的非标准超完备化.
Nonstandard analysis is the new branch to research various mathematical prob-lems using nonstandard models. Since nonstandard analysis is founded by A. Robin-son in1961, the field of real numbers and various relations over it are called the standard model. In the standard model, or in the reals, analysis is said to be stan-dard analysis. And the enlargement of the reals and the relations over it are called the nonstandard model. In the nonstandard model, or in the hyperreals, which is the extension of the reals, analysis is said to be nonstandard analysis. Nonstandard analysis is the true extension of standard analysis. The difference between them is the infinitesimal and the infinite.
The research mainly studies the nonstandard model theory and some applica-tions in fuzzy topology. There are two purposes in this research. One is to convert the nonstandard model into a superstructure, which can convenient for the twice model, even the multi-model. The other is to provide the method for the combi-nation of nonstandard analysis and fuzzy topology. These attempts not only can improve nonstandard analysis, enriching its applications, but also can provide a new method to study fuzzy topology. The main conclusions in this research are as follows:
(1) The individuals set S of a class and a superstore V(S) on it are given, and it is shown that V(S) is large enough. Founded first-order language in the math-ematical logic, the concepts of abstract nonstandard model, specific nonstandard model and pre-nonstandard model are defined with the interpretation mapping in model theory. A new superstructure V(*S) is constructed with ultrapower, and it is proved that V(*S) is a nonstandard model of V(S) by ultrafilter. The necessi-ties of ultrafilter and bounded quantifiers sentence are shown in transfer principle. The structure and properties of mapping*in nonstandard analysis are discussed from several views. Thus, from a superstructure to a new superstructure, it is the foundation of the twice model or the multi-model.
(2) The standard entities, internal entities and external entities in V(*S) are studied. It is shown that*V(S) is only a symbol, which is not the*-image of V(S). The existence ofκ-enlarged model is proved by κ-adequate filter, and the sufficient and necessary conditions and some interesting properties forκ-enlarged model are obtained. Similar to Luxemburg's way, the existence, necessary conditions and some properties ofκ-saturated model are discussed. Based on these, the existence of the twice model V(**S) is proved. Furthermore, the enlargement and saturation for the twice model are discussed with the entities in it, such as the twice standard entities, the twice internal entities, etc. The basic ideas and methods are provided, in this part, for the multi-model in future.
(3) The relations between a complete lattice L and its nonstead extension*L are discussed. The concepts of internal complete lattice, the κ-copy complete lattice and κ-complete lattice are defined. Especially, K-complete lattice is very important for fuzzy topology. For any nonempty subset A, which cardinal less than κ, in a lattice L, if both∨A and∨A exist, L is called κ-complete lattice. Based on it, the representations and properties of the enlarged model and the saturated model are obtained in fuzzy forms. It is a basic way to further research.
(4) The nonstandard fuzzy sets are defined, and some properties of the family of all nonstandard fuzzy sets*[0,1]*x and its subfamilies are discussed, such as standard fuzzy sets°[0,1]x, internal fuzzy sets*[0,1]*x, etc. With standard part mapping st, the relations between fuzzy sets and nonstandard fuzzy sets are shown. Then the nonstandard ultra-compactification (*X, δ) of a fuzzy topological space (X,δ), even Stone-Cech ultra-compactification (*X,δ) are obtained by a nature way.
(5) Based on the interval numbers Ⅱ(R) and its order, the hyperinterval numbers Ⅱ(*R) are provided. And an equivalence relation～is defned in Ns(Ⅱ(*R)), which is subset of Ⅱ(*R), and it is proved that Ⅱ(R) and N(Ⅱ(*R))/～are orderi-somorphic. Then a nonstandard extension (*X,*p) of an interval numbers valued metric space (X, p) is discussed. It is an hyperinterval numbers valued metric space. Furthermore, the completion of (X, p) is obtained by nonstandard points in (*X,*p), finite points Fin(*X) and qusi-near standard points Qns(*X).
(6) The monad m(?) of a fuzzy filter on X X X is defined. And a sufficient and necessary condition is shown when a fuzzy filter on X x X is a fuzzy uniformity on X. It is that m(?) is a fuzzy equivalence relation on*X. The nonstandard ex-tension (*X,(?) of fuzzy uniform space (X,?) and its properties are discussed. It is proved that (*X,?) is a nonstandard fuzzy uniform space, and it is nonstandard ultra-completion of (X,(?)
本论文主要研究了非标准模型的理论及其在模糊拓扑学中的若干应用,论文研究的主要目的有两个,一是尝试着将非标准模型归结为超结构,便于二次模型或多次模型的构造,二是尝试着将非标准分析方法应用于模糊拓扑学中,为非标准分析理论和模糊拓扑学理论的结合提供可能.这些尝试不仅完善了非标准分析理论,丰富了非标准分析方法的应用和研究领域,而且为模糊拓扑学的研究提供了一种新思路和新方法.论文具体研究内容如下：
(1)定义了一个集类上的个体集S及其上超结构V(S),证明了超结构V(S)是一个足够大的集合.以数理逻辑的一阶形式语言为基础,借助模型论中的解释映射提出了抽象非标准模型、具体非标准模型以及准非标准模型的概念.以超滤子为工具,以超幂的构造方法,构造了一个新的超结构V(*S),利用超滤子的特性,证明了新的超结构V(*S)是V(S)的一个非标准模型,指明了超滤子及有界量词句子在转换原理中的必要性.从几个角度说明了非标准分析理论中的*映射的构造,及其保持Boolean运算的性质.这样从一个超结构出发,将非标准模型又归结为了超结构,为进一步讨论二次模型打下了基础.
(2)给出超结构V(*S)中实体的概念,指出标准实体、内实体和外实体的差别,说明了常见记号*V(sS的符号性.定义了k-扩大模型,从k-充足的滤子出发,证明了k-扩大模型的存在性,并以此得到的超结构V(*S)是k-扩大模型的充分且必要条件,以及k-扩大模型的一些有趣的性质.相仿Luxemburg的方法,得到了k-饱和模型的存在性,讨论了k-饱和模型的充分且必要条件,以及k-饱和模型的性质.在这些基础上,证明了二次非标准模型V(**S)的存在性,定义了V(**S)中的一些实体,如二次标准实体、二次内实体等,利用这些实体,讨论了二次非标准模型的扩大性和饱和性.为将来进一步讨论多次非标准模型提供了基本的研究思路和方法.
(3)讨论了完备格L与其非标准扩张*L的关系,定义了内完备格,k-拷贝完备格与k-完备格,特别是k-完备格,对于一个格中的任意非空子集A,若|A|
(4)定义了非标准模糊集合的概念,讨论了全体非标准模糊集合之族*[0,1]*X及其一些子族,如标准模糊集合之族σ[0,1]X)、内模糊集合之族*([0,1]X)等的性质.利用标准部分映射st将非标准模糊集合与模糊集合之间建立了关系.以模糊拓扑空间(X,δ)为基础,讨论了非标准模糊拓扑空间(*X,δ)的超紧性,并给出了模糊拓扑空间(X,δ)一种自然的Stone-Cech超紧化(X,ε)的构造方式.
(5)以区间数集Ⅱ(R)及其上的序关系为基础,定义了超区间数集Ⅱ(*R),并在其子集Ns(Ⅱ(R))上定义了一个等价关系～,证明了Ⅱ(R)与Ns(Ⅱ(*R))/～是序同构的.讨论了区间值度量空间(X,ρ)的非标准扩张(*X,*ρ),证明了(*X,*ρ)是一个超区间值度量空间.研究了(*X,*ρ)的有限点集Fin(*X)和拟近标准点集Qns(*X)两类特殊点集,并以此给出了区间值度量空间(X,ρ)的完备化的一种构造方式.
(6)定义了X×X上的模糊滤子F及其单子m(F),给出了X×X上的一个模糊滤子是X上的模糊一致结构的充分且必要条件,即m(F)是*X上的一个模糊等价关系.讨论了模糊一致空间(X,u)的非标准扩张(*X,u)的性质,证明了(*X,u)是一个模糊一致空间,并且它是(X,u)的非标准超完备化.
Nonstandard analysis is the new branch to research various mathematical prob-lems using nonstandard models. Since nonstandard analysis is founded by A. Robin-son in1961, the field of real numbers and various relations over it are called the standard model. In the standard model, or in the reals, analysis is said to be stan-dard analysis. And the enlargement of the reals and the relations over it are called the nonstandard model. In the nonstandard model, or in the hyperreals, which is the extension of the reals, analysis is said to be nonstandard analysis. Nonstandard analysis is the true extension of standard analysis. The difference between them is the infinitesimal and the infinite.
The research mainly studies the nonstandard model theory and some applica-tions in fuzzy topology. There are two purposes in this research. One is to convert the nonstandard model into a superstructure, which can convenient for the twice model, even the multi-model. The other is to provide the method for the combi-nation of nonstandard analysis and fuzzy topology. These attempts not only can improve nonstandard analysis, enriching its applications, but also can provide a new method to study fuzzy topology. The main conclusions in this research are as follows:
(1) The individuals set S of a class and a superstore V(S) on it are given, and it is shown that V(S) is large enough. Founded first-order language in the math-ematical logic, the concepts of abstract nonstandard model, specific nonstandard model and pre-nonstandard model are defined with the interpretation mapping in model theory. A new superstructure V(*S) is constructed with ultrapower, and it is proved that V(*S) is a nonstandard model of V(S) by ultrafilter. The necessi-ties of ultrafilter and bounded quantifiers sentence are shown in transfer principle. The structure and properties of mapping*in nonstandard analysis are discussed from several views. Thus, from a superstructure to a new superstructure, it is the foundation of the twice model or the multi-model.
(2) The standard entities, internal entities and external entities in V(*S) are studied. It is shown that*V(S) is only a symbol, which is not the*-image of V(S). The existence ofκ-enlarged model is proved by κ-adequate filter, and the sufficient and necessary conditions and some interesting properties forκ-enlarged model are obtained. Similar to Luxemburg's way, the existence, necessary conditions and some properties ofκ-saturated model are discussed. Based on these, the existence of the twice model V(**S) is proved. Furthermore, the enlargement and saturation for the twice model are discussed with the entities in it, such as the twice standard entities, the twice internal entities, etc. The basic ideas and methods are provided, in this part, for the multi-model in future.
(3) The relations between a complete lattice L and its nonstead extension*L are discussed. The concepts of internal complete lattice, the κ-copy complete lattice and κ-complete lattice are defined. Especially, K-complete lattice is very important for fuzzy topology. For any nonempty subset A, which cardinal less than κ, in a lattice L, if both∨A and∨A exist, L is called κ-complete lattice. Based on it, the representations and properties of the enlarged model and the saturated model are obtained in fuzzy forms. It is a basic way to further research.
(4) The nonstandard fuzzy sets are defined, and some properties of the family of all nonstandard fuzzy sets*[0,1]*x and its subfamilies are discussed, such as standard fuzzy sets°[0,1]x, internal fuzzy sets*[0,1]*x, etc. With standard part mapping st, the relations between fuzzy sets and nonstandard fuzzy sets are shown. Then the nonstandard ultra-compactification (*X, δ) of a fuzzy topological space (X,δ), even Stone-Cech ultra-compactification (*X,δ) are obtained by a nature way.
(5) Based on the interval numbers Ⅱ(R) and its order, the hyperinterval numbers Ⅱ(*R) are provided. And an equivalence relation～is defned in Ns(Ⅱ(*R)), which is subset of Ⅱ(*R), and it is proved that Ⅱ(R) and N(Ⅱ(*R))/～are orderi-somorphic. Then a nonstandard extension (*X,*p) of an interval numbers valued metric space (X, p) is discussed. It is an hyperinterval numbers valued metric space. Furthermore, the completion of (X, p) is obtained by nonstandard points in (*X,*p), finite points Fin(*X) and qusi-near standard points Qns(*X).
(6) The monad m(?) of a fuzzy filter on X X X is defined. And a sufficient and necessary condition is shown when a fuzzy filter on X x X is a fuzzy uniformity on X. It is that m(?) is a fuzzy equivalence relation on*X. The nonstandard ex-tension (*X,(?) of fuzzy uniform space (X,?) and its properties are discussed. It is proved that (*X,?) is a nonstandard fuzzy uniform space, and it is nonstandard ultra-completion of (X,(?)
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