【作者】 袁学海；

【导师】 夏尊铨；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2006， 博士

【摘要】 首先，利用模糊点与模糊集的邻属关系，给出了（（?），（?））-模糊映射，（（?），（?））-凸模糊锥和（（?），（?））-模糊拓扑的定义，其次，研究了模糊集的范畴，凸模糊锥的范畴和合意集的范畴，给出了中间元和弱topos的定义。弱topos理论是介于卡氏积封闭范畴和topos之间的一种理论，它有类似于topos理论的功能。最后，在topos中引入模糊子对象的概念，将Zadeh的模糊子集的概念推广到了topos中。具体研究工作如下： 1．在第2章中，首先引入了（（?），（?））-模糊映射的定义，并将（∈，∈）-模糊映射，（∈，∈∨q）-模糊映射和（（?），（?）∨（?））-模糊映射推广为(λ，μ]-模糊映射，研究了(λ，μ]-模糊映射与HX-模糊映射的关系。其次，以模糊集为对象，以(λ，μ]-模糊映射为态射，建立了范畴Fuz_λ^μ。证明了范畴Fuz_λ^μ，实值模糊集的范畴RVF，从小范畴C到范畴Fuz的函数范畴Fuz^C为一个弱topos。我们研究了弱topos的性质，揭示了两个对象的最小特征之间的关系，单态射f∶A′→A的特征χ_f与A′的最小特征α_A′，之间的关系，给出了中间元m∶Λ→Δ中的Λ为最终元的充要条件，并对一个对象的“幂元”做了刻画。最后，在topos中给出了一个对象的模糊子对象的定义，将Zadeh的模糊子集的概念推广到了topos中，建立了模糊子对象的范畴FC，证明了范畴FC是有限完备的。 2．在第3章中，首先引入了（（?），（?））-凸模糊锥的定义，得到了（∈，∈）-凸模糊锥，（∈，∈∨q）-凸模糊锥和（（?），（?）∨（?））-凸模糊锥。其次，利用合意空间理论，给出了C-凸模糊锥的定义，证明了（∈，∈）-凸模糊锥为C-凸模糊锥，每一个C-凸模糊锥都同构于一个凸锥生成的C-凸模糊锥。再次，建立了凸模糊锥的范畴CFC，证明了范畴CFC是有限完备的且有类似的Exponential性质。最后，建立了合意集的范畴C（Ω，（?）），证明了范畴C（Ω，（?））为一个topos。 3．在第4章中，首先引入了（（?），（?））-模糊拓扑的概念，得到了（∈，∈）-模糊拓扑，（∈，∈∨q）-模糊拓扑和（（?），（?）∨（?））-模糊拓扑。并将这三种模糊拓扑推广为(λ，μ]-模糊拓扑。其次，给出了基于模糊逻辑蕴涵算子R的R-模糊拓扑的概念，这是应明生的模糊化拓扑的推广。证明了（∈，∈∨q）-模糊拓扑为R_G-模糊拓扑，（（?），（?）∨（?））-模糊拓扑为（?）-模糊拓扑。最后，利用合意空间理论，给出了C-模糊拓扑的定义，证明了（∈，∈）-模糊拓扑为C-模糊拓扑，并研究了C-模糊拓扑的性质。更多还原

【Abstract】 Firstly, by the use of the relations between fuzzy points and fuzzy sets, the definitions of （β|-,α|-）-fuzzy mapping, （β|-,α|-）-convex fuzzy cone and （β|-,α|-）-fuzzy topology are introduced. Secondly, the category of fuzzy sets, category of convex fuzzy cone and category of consensus set are studied respectively. The concepts of middle object and weak topos are acquired. Weak topos is a new kind of categorical theory which is stronger than Cartesian colsed category and weaker than topos theory, and a weak topos can serve a similiar function to a topos. Finally, the concept of fuzzy subobject is given and the concept of Zadeh’s fuzzy subset is generalized to a topos. The main results obtained in this dissertation are as follows:1. In Chapter 2, first, the definition of （β|-,α|-）-fuzzy mapping is introduced, three fuzzy mappings such as （∈,∈）-fuzzy mapping, （∈,∈∨ q）-fuzzy mapping and （∈|-,∈|- ∨ q|-）-fuzzy mapping are obtained. By generalizing those three fuzzy mappings, (λ,μ]-fuzzy mapping is acquired and the relations between (λ,μ]-fuzzy mapping and HX-fuzzy mapping are discussed. Second, a category Fuz_λ^μ of fuzzy subsets and (λ,μ]-fuzzy mappings is built. It is proved that the category Fuz_λ^μ is a Cartesian closed category, but the category Fuz_λ^μ is not a topos. Third, the concept of middle object and weak topos are introduced. It is proved that the category Fuz_λ^μ, the category RVF of real valued fuzzy sets and the category Fuz^C of functors from small category C to the category Fuz are weak topos. The properties of a weak topos are studied. The relations between the smallest characteristic morphisms of two objects and the relation between the characteristic morphism X_f of monomorphism f : A’ → A and the smallest characteristic morphism α_A’ of object A’ are described. A sufficient and necessary condition that A is a Terminal object for middle object m : Λ → Δ is acquired and power object of an object is described. Final, the concept of fuzzy subobject is given and the concept of Zadeh’s fuzzy subset is generalized to a topos. The category FC of fuzzy subobject is built and it is proved that the category FC is finitely complete.2. In Chapter 3, first, the concept of （β|-,α|-）-convex fuzzy cone is given and （∈,∈）-convex fuzzy cone, （∈,∈∨ q）-convex fuzzy cone and （∈|-,∈|- ∨ q|-）-convex fuzzy cone are obtained. Second, by the use of consensus space, the definition of C-convex fuzzy cone is given. It is proved that a （∈,∈）-convex fuzzy cone is a C-convex fuzzy cone, and a C-convex fuzzy cone is isomorphic to the C-convex fuzzy cone generated by a cone S. Third, the category CFC of convex fuzzy cone is built. It is proved that the category CFC is finitely complete and has the similar properties to Exponential. Final, the category C（Ω,（?）） of consensus sets is built and it is proved that the category C（Ω,（?））is a topos.更多还原