Matroid of Covering Fuzzy Rough Sets

Rough set, which proposed by Z.Pawlak firstly, is an effective and efficient tools to deal with uncertain, vagueness and granular ininformation system through indiscernibility relation.It is widely used in some area of artificial intelligence, data mining and knowledge discovery.However, fuzzy set theory, is proposed by Zadeh who is the America computers and cybernetics expert in1965, is a mathematical theory to deal with fuzziness and fuzzy concept. And it can be combined with fuzzy control, fuzzy recognition, fuzzy reasoning, and fuzzy decision, which used in complex system, impoves the system performance. But, the set of Pawlak is precise. And we often involve with many fuzzy and uncertain concept. Those two theories are close relationship, and complementary is strong.So, Dubois and Prade propose the concepts of fuzzy rough set and rough fuzzy set, which is an extention of rough set.Combined with fuzzy set, we can effectively and efficiently deal with information.Matroid is an important branch of combinatorial mathemetics.Not only is the theory structure more perfectly, but it is widely used in many technology fileds,such as integer programming, electronic network.The problems of operational research such as network flow and combinatorial optimization such as minimum spanning tree can be extended into matroid.The concept of matroid is an extention of combinatorial mathemetics and algebra.In this paper, we extend equivalence relation to partially ordered relation, and we give some definitions and discuss some properties of covering rough set on partially ordered set.On one hand, we give some models of rough set and the upper and lower approximations definition of rough set on partially ordered sets.And we get some properties from the up definition.And then, we introduce covering-based reduction algorithm and partly improve this algorithm.we propose the reducible element on partially ordered set of rough set.On the other hand, we extend the covering fuzzy rough set theory and combine the matroid theory. Then, we define approximation space of covering-based fuzzy rough set on matroid.And we give the definition of upper and lower approximations of covering-based fuzzy rough set on matroid.Therefore; this is an extention of rough set to matroid field.From the point view of numerical character, we analyze the properties of rough equivalence.Then, we define the concepts of precision approximation and rough degree.We can combine rough set theory with some mathematics tools such as linear algebra and graph theory to form space. It is great helpful to apply rough set to data mining.At present,the analysis and study of combining rough set with matroid is still in its beginning stages,it is looking forward to the development prospects of rough set theory.