Distributive Equations Of Implications Based On Ontinuous Triangular Morns

In this paper, we completely characterize the solutions for the distributive equations ofimplications based on a continuous t-norm（or t-conorm）. Specifically speaking, two topics arediscussed: one is the solutions of distributive equations of implications based on continuous t-norms, the other is the solutions of the other two distributive equations of implications basedon continuous t-conorms.In Chapter1, conception and properties of t-norms, t-conorms, and fuzzy implications arereviewed. Variety and construction of t-norms, t-conorms are introduced, the relations of contin-uous t-norms, t-conorms and additive generator are given, finally, fuzzy implications’definitionand the four similar to the additive Cauchy functional equations’properties are recalled.In Chapter2, we explore the solution of functional equations I（T （x, y）, z）=S（I（x, z）, I（y, z））,where T:[0,1]²â†'[0,1] is a continuous t-norm, S:[0,1]²â†'[0,1] is a continuous andarchimedean t-conorm, I:[0,1]2â†'[0,1] is an unknown function. Under the assumptionsthat I is continuous except the horizontal section I（x,1）=1for x∈[b,1], we get its completecharacterizations.In Chapter3, we explore the solution of functional equations I（x, S₁（y, z））=S₂（I（x, y）, I（x, z））,where S₁:[0,1]²â†'[0,1] is a continuous t-conorm, S₂:[0,1]²â†'[0,1] is a continuous andarchimedean t-conorm, I:[0,1]²â†'[0,1] is an unknown function. Under the assumptions that Iis continuous except the section I（0, y）=1for y∈[0, a], we get its complete characterizations.