The Full Characterizations Of The Solutions For The Distributivity Of Fuzzy Implication Based On Continuous T-conorm

In order to avoid combinatorial rule explosion in fuzzy reasoning, Qin and Baczynski[26]investigated the distributive equation of implication I(x, T1(y, z))=T2(I(x, y), I(x, z)) and ob-tained all solutions of I when T1is continuous but not Archimedean, T2is continuous andArchimedean. In this thesis, we continue to explore and give out the sufcient and necessary con-ditions of solutions for the distributive equation of implication I(x, S1(y, z))=S2(I(x, y), I(x, z)),when S1and S2are continuous but not Archimedean triangular conorms, and I is an unknownfunction.Specially, the work is listed in the following.In chapter1, definitions and properties of continuous t-conorm, continuous Archimedeant-conorm and some results pertaining to the addition Cauchy functional equation are recalled.In chapter2, to characterize all solution of Eq.(4), we introduce several lemmas and provemonotonicity of I(x,.) and property of preserving idempotent element of I(x,.). Besides, weprove that the inverse image of I(x,.) under a generating interval is yet a generating interval.In chapter3, we consider all solutions of Eq.(4) when a generating triangular conorm ofS1on the open generating subinterval (am, bm) is strict and a generating triangular conorm ofS2on the open generating subinterval (cj, dj) is strict or nilpotent. Further, I(x,.) is uniquelydetermined up to a positive multiplicative constant.In chapter4, we consider all solutions of Eq.(4) when a generating triangular conorm of S1on the open generating subinterval (am, bm) is nilpotent and a generating triangular conorm ofS2on the open generating subinterval (cj, dj) is strict or nilpotent. Further, I(x,.) is uniquelydetermined up to a positive multiplicative constant.