Estimation Of The Idempotent Index Based On AFS Structures

Fuzzy mathematics is a newly subject. It develops very quickly, since the concept of fuzzy set and the first announced article was proposed by Professor L.A.Zadeh, an American cyberneticist, in 1965. In the recent 40 years, fuzzy mathematics has already been used in national economy, science technique and other realms. The AFS (the Axiomatic Fuzzy Sets) theory was firstly put forward by Professor Liu Xiaodong in 1995. It applies the new mathematics object AFS algebras, which is a non-Boolean algebra. The AFS structure is a kind of special "system" and cognitive field, as to the representation problem of relation function to carry on the research, establish a new fuzzy logic system using topological molecular lattices. It depicts abstract relation between human conceptions. The establishment of relation function and the fuzzy logic systems are objectivity, rigor and unity. Currently the AFS theory has been applied in the many realms successfully, such as cluster analysis, pattern recognition and hitch diagnosis etc.The AFS theory gives a new way on the pattern recognition and hitch diagnosis of the complicated system lately, when we create the AFS structure (Y,T,X) according to the expertise, The matrix Mτ can not include enough information because of various reason, in order to add the lost information, we must find the matrix Mτ,which satisfy M^2t_τ = M^r_τ, then ( Y, r , X) can be made to be a AFS structure . M_2rτ< M^r_τ always establishes, so the M^r_τ is the complement of the information to the M_τ. By creating structure r with the M^r_τ ,it can make the information of new AFS structure integrity. There is a homomorphism relation between Matrix M_τ and B_n. Thus some conclusions of the Boolean matrix can be applied to find r, which satisfy M^2r_τ = M^r_τ, so the process of operation of the computer can be simplified and the accuracy of the results is easy to be verified.In this paper, we give the way of the estimation of the idempotent index and introduce the concept of Boolean Matrix on the set M, and discuss some of its correlatedproperties of transitive closure. We also present a homomorphic mapping between the Boolean Matrix on the set M and (0-1) Boolean Matrix, and prove this theorem. Finally we give the result about the estimation of integer r. So its strong objectivity can be easily used in process of data mining, and reduce complexity in theoretical studies.