Granular Computing Theory Based on Hypergroups
Yuan Xue-hai, Hongxing Li, Kaibiao Sun · Mohu xitong yu shuxue · 2011
This paper is first article to apply the theory of hypergroup in algebra into the research of granular computing.Firstly,we introduce the definitions of normal hypergroup and strong normal hypergroup,and show that each normal hypergroup can be generated by a strong normal hypergroup.Then by choosing T in the quotient space theory model(X,f,T) of granular computing as hypergroup structure,we prove that x and y are in the same path in the model(X,f,T) if and only if [x] and [y] are in the same path in quotient space model(,[T]) by the use of homomorphism of hypergroup;Further,we prove that if X and Y are homomorphism of hypergroup,then the quotient space induced by them are also homomorphism.Secondly,we develop the relationships between normal hypergroup and some theories such as ample field in possibility theory,Pawlak approximation space and topological space.We point out that(1) Strong normal hypergroup and ample field are equivalent;(2) Strong normal hypergroup and Pawlak approximation space are equivalent;(3) The upper and lower approximations of sets can be defined by hypergroup,and the homomorphism of hypergroup can be described by the upper and lower approximations;(4) Strong normal hypergroup can be generated by topological space,and normal hypergroup can be generated by the strong normal hypergroup generated by topological space;(5) The ample field in possibility theory and Pawlak approximation space are equivalent,and the ample field is just the set of all well defined sets.Hence,the theory of hypergroup can be applied to the research of granular computing.