基于三角模的模糊拓扑向量空间研究

项目名称： 基于三角模的模糊拓扑向量空间研究

项目编号： No.11301281

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 张化朋

作者单位： 南京邮电大学

项目金额： 22万元

中文摘要： 模糊拓扑向量空间是以有着深刻实际背景的模糊集合论为基础建立起来的一种使模糊拓扑结构和代数结构（线性结构）在同一载体上得到协调交融的空间结构，而三角模是定义在单位区间 [0,1] 上的幺半群。本项目致力于将三角模与模糊拓扑向量空间的研究有机地结合，拟建立基于三角模的模糊拓扑向量空间理论。其主要研究内容有：(1) 提出基于三角模的模糊拓扑向量空间的概念，给出其分别借助于局部基和模糊伪范数族的刻画；(2) 研究该空间的两类重要的子空间，即局部凸空间和局部有界空间，分别利用一族模糊半范数和模糊准范数来刻画这两类空间；(3) 引入模糊范数的概念，建立该空间的可赋范化定理。 本项目研究的课题属于多学科交叉的前沿。开展本项目的研究，对于进一步丰富和发展模糊拓扑向量空间理论，扩大三角模的应用范围，促进一般拓扑学、代数学和模糊数学等学科的相互渗透和发展具有重要的理论意义和学术价值。

中文关键词： 模糊拓扑；模糊拓扑向量空间；三角模；局部凸性；概率范数

英文摘要： Fuzzy topological vector space (FTVS) is established based on fuzzy set theory which has deeply practical backgroud and it is a space structure which harmonizes and blends fuzzy topological structure with algebraic structure (linear structure) on the same carrier, while triangular norm (t-norm) is a monoid defined on the unit interval [0,1]. This project is devoted to combining t-norm with FTVS organically and establishing the theory of FTVS based on t-norm. The contents of this project are as follows: (1) the notion of FTVS based on t-norm will be proposed and it will be characterized in terms of local base and a family of fuzzy pseudo-norms, respectively. (2) two important kinds of subspaces, i.e., locally convex space and locally bounded space will be investigated and they will be characterized by a family of fuzzy semi-norms and a family of fuzzy quasi-norms, respectively. (3) the concept of fuzzy norm will be proposed and the normability theorem for the space will be established. The subject of this project is interdisciplinary. The research of this project has important theoretical significance and academic value in further enriching and developing the theory of FTVS, enlarging the application range of t-norms, and promoting mutual saturation and development of general topology, algebra and fuzzy mathemati

英文关键词： Fuzzy topology；Fuzzy topological vector space；Triangular norm；Local convexity；Probabilistic norm