The Matroidal And Topological Structure Of Multigranulation Covering Rough Set

Rough set theory is a useful mathematical tool for dealing with the incompleteness dataand uncertainty data, it has wide application in artificial intelligence and data mining, etc.Covering rough set theory is an extension of classical rough set theory, in which everycovering is regarded as a granulation. However, there are some limitations on dealing withsome practical issues. Based on the depiction of the minimal description in covering rough set,first, we define the minimal description of multigranulation. Then we construct a coveringrough set model. Next we give the upper and lower approximation operators and investigatesome properties of them. Finally, we investigate the attribute reduction based on the minimaldescription of multigranulation.It possesses important theory significance and practical value to combine rough set withother theories. As a new research field, matroid theory is the abstraction and extension ofgraph theory and algebra concept, it has wide application in combinatorial optimization,integer programming, network flow and grid theory. In this paper, we propose thecombination of rough set, matroid and topology.The main content of the paper is stated asfollows:1. We define a multigranulation matroid and a multigranulation rank function. Based onwhich we describe the multigranulation matroid approximation operators and discuss theproperties of them. Then we investigate the relationship between the multigranulation matroidapproximation operators and the multigranulation rough set approximation operators. Basedon the multigranulantion matroid, we propose a dual multigranulation matroid. Furthermore,we construct a rough set model by the dual multigranulation matroid.2. Both topology and matroid theory have concepts like base, closed set and closure, butthe concepts in different theory are defined in different ways. In this paper we define atopological space by the closure of matroid first, and use the closure to depict the separationproperty of topological space, discuss the connection among different separation properties indetail, then investigate the relationship between the closure of topology and the closure of matroids, construct a matroid based on topological structure by this relationship. In particular,we consider a class of matroid constructed by the so called upper approximation numbers.The matroids are closely related to rough set. Therefore, we give an explicit characterizationof the separation property of the topological spaces induced by these matroids.