粗代数研究

On Rough Algebras

在粗糙集的代数方法研究中,一个重要的方面是从粗糙集的偶序对(?下近似集,上近似集?)表示入手,通过定义偶序对的基本运算,从而构造出相应粗代数,并寻找能够抽象刻画偶序对性质的一般代数结构.其中最有影响的粗代数分别是粗双Stone代数、粗Nelson代数和近似空间代数,它们对应的一般代数结构分别是正则双Stone代数、半简单Nelson代数和预粗代数.通过建立这些粗代数中算子之间的联系,证明了:(a)近似空间代数可转化为半简单Nelson代数和正则双Stone代数;(b)粗Nelson代数可转化为预粗代数和正则双Stone代数;(c)粗双Stone代数可化为预粗代数和半简单Nelson代数,从而将3个不同角度的研究统一了起来.

Description of the pairs ?low approximation, upper approximation? of rough sets is an important aspect in the research of rough set theory by algebraic method. By defining some basic operators on the approximation pairs, rough algebras can be constructed. Then some general algebras can be selected to describe the pairs of rough sets. The most famous rough algebras are Rough Double Stone Algebra, Rough Nelson Algebra and Approximation Space Algebra, and their corresponding general algebra structures are regular double Stone algebra, semi-simple Nelson algebra and pre-rough algebra respectively. This paper establishes the relations between the operators of these rough algebras and proves that: (a) approximation space algebra can be made into semi-simple Nelson algebra or regular double Stone algebra; (b) rough Nelson algebra can be made into pre-rough algebra or regular double Stone algebra; (c) rough double Stone algebra can be made into pre-rough algebra or semi-simple Nelson algebra. Thus, a uniform structure for the famous works from three different aspects is built and the relations among them are established.