基于Quantale理论的粗糙集代数与拓扑性质

In order to study the generalized rough sets, which based on complete boolean lattice, complete completely distributive lattice and complete residuated lattice, in a unified framework, the generalized approximation operators on a quantale are introduced in this project. The main aim is to research the algebraic and topological properties of generalized rough set based on quantale theory. Firstly, we construct the upper and lower approximations based on binary relation and covering respectively, the algebraic properties and lattice structures of the collections will be discuss, the lower and upper approximation operators are described with axiom set. Secondly, considering the case that the upper and lower approximation operators are the closure and interior operators of a topology, as well as quantic nucleus and quantic conucleus. We prove that there exists a one-to –one conrrespondence between the set of some kind of binary relations and the set of topologies which satisfy some conditions. We investigate the topological properties of the topological space which induced by the rough sets on quantale . And the topological properties for topological spaces are extended to generalized approximation spaces of quantale. Finally, the lower and upper approximation are constructed based on congruence relations determined by ideals of a quantale, the algebraic properties will be discuss.

为使得基于完备布尔格、完备的完全分配格和完备剩余格的各类广义逼近算子统一在同一框架下进行研究，本项目提出了Quantale上的广义逼近算子，主要目的是研究基于Quantale理论的广义粗糙集的代数与拓扑性质。首先，我们利用Quantale上的二元关系和覆盖分别构造上下近似算子，研究其基本代数性质和各种集合的格结构，并用公理集对上下近似算子进行刻画。其次，考虑Quantale上的上下近似算子构成拓扑的闭包和内部算子的情形，以及构成核映射与预核映射情形，得到某种二元关系与满足一定条件拓扑的1-1对应关系，研究Quantale上的粗糙集所诱导的拓扑空间的拓扑性质，将一般拓扑空间的拓扑性质延伸到Quantale上的广义逼近空间。最后，对Quantale上由理想所确定的同余关系构造的上下近似算子的代数性质进行研究。

为使得基于完备布尔格、完备的完全分配格和完备剩余格的各类广义逼近算子统一在同一框架下进行研究, 本项目研究了Quantale上的广义逼近算子。首先,对Quantale上由理想所确定的同余关系构造的上下近似算子的代数性质进行了研究，并在Quantale 研究了导子的相关性质，以及导子与核(余核)映射之间的关系。第二，我们利用Quantale上的二元关系构造上下近似算子, 研究了其基本代数性质和各种集合的格结构,并用公理集对上下近似算子进行了刻画。第三, 考虑Quantale上的上下近似算子构成拓扑的闭包和内部算子的情形,以及构成核映射与预核映射情形,得到某种二元关系与满足一定条件拓扑的1-1对应关系。最后，我们利用粗糙概念、逼近概念等去构造信息系统的一个任意对象集（属性集）的上、下逼近，在两种不同算子情形下对其性质进行了研究。粗糙集和格理论的交叉研究，有助于推动理论计算机科学的发展,也使得其自身的研究内容越来越丰富。