【作者】 张家锋；

【导师】 秦克云；

【作者基本信息】 西南交通大学 ， 应用数学， 2006， 硕士

【摘要】 粗糙集理论是波兰科学家Z. Pawlak于1982年提出的一种数据分析理论，目前已发展成为一种处理模糊和不确定性信息的数学理论，井且成功地应用于机器学习、模式识别、决策支持、数据挖掘、过程控制等领域。粗糙集理论在数据库知识发现中的应用成功推动了粗糙集理论的研究，粗糙集的代数结构分析是粗糙集理论研究中最活跃的研究分支之一。由于粗糙集代数具有基本的逻辑代数结构，若能建立粗糙集代数和逻辑代数的联系，就可以借助已有的逻辑系统的研究结果来讨论粗糙逻辑并深入研究粗糙集的结构。本文研究粗糙集代数与非经典逻辑代数的关系，将粗糙集理论应用于MV-代数和R₀-代数，讨论其滤子的粗糙性并研究同态映射之下滤子的性质。主要取得了如下的研究结果：1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，从而构造出相应的粗代数，进而找到能够抽象刻画偶序对性质的一般代数结构，比如剩余格，BL-代数，MV-代数，R₀-代数。2滤子是非经典逻辑代数中的一个基本结构，对于相应逻辑系统的研究具有重要意义。本文将粗糙集理论应用于滤子理论，引入了上、下粗糙滤子的概念，讨论它的基本性质，推广了非经典逻辑代数中滤子的相关性质。3对于R₀-代数上的同态映射f，引入了f的对偶核的概念，并证明了对偶核是一个滤子。更多还原

【Abstract】 Rough set theory is a new theory of data analysis, it was first put forward by Poland scientist Z.Pawlak. At present it has been developed to be a new mathematical tool to deal with vagueness and uncertainty. It has been applied to many areas successfully including machine learning, pattern recognition, decision support, data mining and process control.The successful application of rough set theory to knowledge discovery in database promotes the study of rough set theory. Algebraic analysis of rough set is one of the most active branches in the research of rough set theory. If the relation between rough algebra and logic algebra was given, we could research the rough logic using the conclusion of logic system and study the structure of rough set.This paper studies the relationship between rough set algebra and non-classical logic algebra, and applys rough set theory to MV-algebra and R₀-algebra. The roughness of filter is studied. The main conclusions are as following:1. Based on the description of the pairs （<low approximation, upper approximation>） of rough set, we define some basic operators on the approximation pairs and construct some rough algebras. Some general algebras structure were selected to describle the pairs of rough set, for exarnple, residuated lattice, BL-algebra,MV-algebra and R₀-algebra.2. In non-classical logic algebras filter is a basic structure and it plays an important role in the research of relative logic system. In this paper, the notions of upper and lower rough filters were introduced, with their basic properties being discussed.3. For homomorphism of R₀-algebra, we introduce the notion of dual kernel and have proved that each dual kernel is a filter.更多还原