利用分子格对粗糙集理论进行推广

Generalization of Rough Set Theory Using Molecular Lattices

粗糙集模型的推广是粗糙集理论研究的重要内容 ,该文将分子格引入到粗糙集理论中作为基本代数系统 ,在分子格中定义了一个从分子到一般元素的映射 ,并通过该映射定义了更为一般和抽象的下近似算子 和上近似算子 ▲ .文中还研究了映射为一般映射、外展映射、对称映射和内缩映射时所定义近似算子的性质 .当映射为一般映射时得到的性质表明文中分子格基础上构造的代数系统 (L ,∧ ,∨ , ,▲ ,0 ,1)是粗糙集代数 (2 U,∩ ,∪ ,～ ,L ,H)的抽象和推广 ;而当映射为外展映射、对称映射和内缩映射时 ,得到了分别与模态逻辑中公理D、公理B和公理 4及它们的对偶相对应的性质 .

Generalization of rough set model is one important aspect of rough set theory study, and it is very helpful to consummate rough set theory. Developing rough set theory using algebra system has been paid great attention, and some researchers reported the significant development in the area. But the base algebra systems, on which approximation operators are defined, are confined to Boolean algebra or special Boolean algebras, including set algebra, Nelson algebra (quasi-pseudo Boolean algebra), and atomic Boolean lattice. This paper introduces molecular lattice as base algebra system on which a map from molecule to general element is defined. Consequently, the lower approximation operator and upper approximation operator are defined using molecules and the map based on the frame of molecular lattice. The approximation operators are more general and abstract compared with approximation operators reported in some papers. The characteristics show that the algebra system (L, ∧, ∨, , , 0,1) based on molecular lattice is one of abstract and generalized forms of rough set algebra(2U,∩ ∪ ∼, L, H) when the map is arbitrary. Characteristics responding to Axiom D, B and 4 (and their dual axioms) in modal logic are obtained respectively when the map is extensive, symmetric and intensive.