Distributive Equations Of Implications Based On Triangular Norms

In this paper, we study the solutions of distributive equations of implications, both inde-pendently and along with other equations. Specifically speaking, two topics are discussed: oneis the solutions of distributive equations of implications based on strict t-norms, both indepen-dently and along with other equations; the other is the solutions of the distributive equations ofimplications based on nilpotent t-norms, both independently and along with other equations.In Chapter 1. Definition and properties of t-norms and fuzzy implication operators arereviewed. Variety and construction of t-norms are introduced. In particular, the properties ofstrict and nilpotent t-norms are given. Finally, several kinds of fuzzy implication operators arerecalled.In Chapter 2. We explore the solution of functional equations I(x,T(y,z)) = T(I(x,y),I(x,z))and I(x,y) = I(N(y),N(x)), where T is a strict t-norm, I a fuzzy implication and N a strongnegation. Under the assumptions that I is continuous except the points (0,0) and (1,1), we getthe full characterizations of the solutions for both functional equations.In Chapter 3. we explore the distributive equations of implications, both independently andalong with other equations. In detail, we consider three classes of equations. (1) By means of thesection of I, we give out the su?cient and necessary conditions of the solutions for distributiveequations of implications I(x,T(y,z)) = T(I(x,y),I(x,z)) based on nilpotent triangular norms,which indicate that there are no continuous solutions for it satisfying the boundary conditionsof implications; Under the assumptions that I is continuous except the vertical section I(0,y),y∈[0,1), we get the full characterizations of the solutions to these equations. (2) We prove thatthere are no solutions for functional equations I(x,T(y,z)) = T(I(x,y),I(x,z)),I(x,I(y,z)) =I(T(x,y),z). (3) We obtain the su?cient and necessary conditions on T and I to be solutionsof functional equations I(x,T(y,z)) = T(I(x,y),I(x,z)),I(x,y) = I(N(y),N(x)).