Distributive Equations Of Trigonometric And Trigonometric Comodulus Based On Fuzzy Implication Of Continuous Triangular Modulo Powers

Distributivity equations of fuzzy implications over triangular norms and triangular conorms are important research topic in the field of fuzzy implications.They play a key role in eliminating combinatorial explosion of rules,studying the algebraic structure of logic algebra and characterizing the analytical structure of fuzzy implications operators.A great deal of researches have been done on the distributivity equations of fuzzy implications over triangular norms and triangular conorms.Recently,S.Massantet et al.proposed a new kind of fuzzy implications based on powers of continuous t-norms,called T-power based implications and made an indepth study of its properties.If the fuzzy implication I satisfies(NP),then I satisfies the four distributivity equations if and only if T=TM,S=SM.f-implications,gimplications,(S,N)-implications,(Q,L)-implications and R-implications all satisfy(NP),but T-power based implications are different from the other well-known classes of fuzzy implications,they do not satisfy the important property(NP).Therefore,the method to prove the necessary and sufficient conditions for the four distributivity equations of T-power based implications over triangular norms and triangular conorms needs another way.The second part in this paper,first of all,T-power based implications are divided into three categories:T-power based implications generated by minimum triangular norms,T-power based implications generated by Archimedean triangular norms as well as T-power based implications generated by ordinal sum triangular norms,in which the T-power based implications generated by Archimedean triangular norms are divided into the T-power based implications generated by product triangular norms and the T-power based implications generated by Lukasiewicz triangular norms.Then four distributivity equations of T-power based implications over triangular norms and triangular conorms are separately studied by classification discussion,necessary and suficient conditions for the four distributivity equations of T-power based implications over triangular norms and triangular conorms are given.Finally,it can be seen from the fact that T-power based implications do not satisfy(EP),for any triangular norms,the T-power based implications do not satisfy the input law.The third part in this paper proves that T-power based implications are new kind of fuzzy implications,they are different from the other well-known classes of fuzzy implications such as f-implications,g-implications,(S,N)-implications,(Q,L)-implications and R-implications,fuzzy implications classification framework are improved.