模糊Domain中的一些范畴之间的对偶等价

项目名称： 模糊Domain中的一些范畴之间的对偶等价

项目编号： No.11201112

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 姚卫

作者单位： 河北科技大学

项目金额： 23万元

中文摘要： 在经典Domain理论中存在着许多对偶等价范畴，如：（1）Locale范畴和拓扑空间范畴之间的Isbell伴随函子可以导出Sober空间范畴和空间式Frame范畴之间的对偶等价性；（2）所有偏序集和所有具有左伴随（相应地，右伴随）的保序映射构成的两个范畴对偶等价;（3）代数Domain范畴和偏序集范畴对偶等价；（4）完全分配格范畴和Domain范畴对偶等价。量化Domain可以看作经典Domain的模糊形式，其中一些范畴间的对偶等价性是一个非常有意义的课题。上述第一组范畴的对偶等价性在量化Domain中的对应我们已用模糊集的方法完成并发表，本项目拟继续研究上述另外三组范畴间的对偶等价性在量化Domain中的对应形式。这些内容的研究可以使我们认识到相关范畴的本质，更关系到量化Domain的模糊集方法的可行性和将来在理论计算机科学中应用的可能性，具有重要意义。

中文关键词： 模糊Domain；(对偶)等价；范畴同构；模糊Scott拓扑；

英文摘要： In classical domain theory, there are many pairs of dually equivalent categoryies, for example: (1) The Isbell adjoint functors between the category of locales (the opposite of the category of frames) and the category of topological spaces can deduce the dual equivalence between the category of Sober spaces and the caetgory of frames. (2) The category of all posets and all monotone maps with left adjonts and the category of all posets and all monotone maps right adjoints are dually equivalent. (3) The category of algebraic domains is dually equivalent to the category of posets. (4) The category of completely distributive lattices is dually equivalent to the category of domains (i.e., continuous dcpos). In some sense, quantitative domains can be considered as a kind of fuzzifications of classical domains. It is very interested and important to investigate the (dual) equivalence between certain categories. The first dual equvalence listed above has been studied by fuzzy approach in our published papers. The aim of this project is to continuous to study the other three duall equivalence in quantitative domains. By these work, we can know the essence of the correponding categories. And it is interested and important for applications of fuzzy-set approach to quantitative domains in theoretical computer science.

英文关键词： fuzzy domain；(dual) equivalence；categorical isomorphism；fuzzy Scott topology；