Matroids Deduced From The Upper Approximation Operators Of Rough Sets

In 1935, further abstracting the concept of interdependency in linear algebra and in graph theory, the mathematician H. Whitney initiated matroid theory, a new branch of mathematics. One of the development characteristics of matroid theory is that it can combine with other subjects very well, such as graph theory, lattice theory, geometry, combination theory and algebra. It equips the theory with many growth points, thus making it developed continuously.In 1982, Polish scholar Z. Pawlak proposed rough set theory, which is a mathematical tool to depict incompleteness and uncertainty. It can effectively analysis all types of inaccurate, inconsistent and incomplete information. Also, it can be used to analyze and infer the data in order to discover the underlying knowledge and potential rules. Now rough sets are not only perfected from the aspect of mathematical theory, but also applied successfully to other fields, such as machine learning, pattern recognition, decision analysis, medical diagnosis, approximate reasoning, process control, image processing, knowledge discovery in database, expert system and so on.Recent years, the combination of matroid theory and rough set theory has attracted many scholars’ interest and made some achievements in both aspect of theory and application. It is a new research field with good development prospects and potential.Based on the upper approximation operators of rough sets, this dissertation constructs one type of matroids and studies its properties. Many related issues arisen from it will be discussed as well.Chapter 1 is introduction. In Chapter 2, we introduce some elementary knowledge of matroids and rough sets. And in Chapter 3, we construct one type of matroids, called the upper approximation operator matroids, through the upper approximation operators of rough sets. Then we investigate the basic properties of this type of matroids. Furthermore, we investigate some special properties of it, such as the sufficient and necessary conditions for different rough sets to induce the same upper approximation operator matroid. In Chapter 4, we study the dual matroids of the upper approximation operator matroids, with which the upper approximation operator matroids called collectively approximation operator matroids. Chapter 5 further discusses the approximation operator matroids. In Chapter 6, we abstract augmentation uniqueness and union minimality from approximation operator matroids and study these two properties in depth.