The Representations Of Trianglar Norms And Their Applications On Solving Fuzzy Relational Equations

This paper deals with the representations of t-norm T over interval valued fuzzy sets and the applications of Archimedean t-norm's additive generators on solving fuzzy relational equations.It first gives a generalization of the representation theorem of Archimedean t-norm T,and proves that Archimedean t-norm's additive generators form a semigroup.After that,it shows a representation theorem of a t-norm over interval valued fuzzy sets.Then this paper investigates the solution sets of fuzzy relational equations by applying Archimedean t-norm's additive generators.It obtains a sufficient and necessary condition that the solution set of fuzzy relational equation sup_i?nT(a_i,x_i)= b is nonempty and a condition that there exists a minimal element in the solution set.Then it verifies the condition that every solution has a minimal one when the solution set is nonempty and describes the solution set.As for the solution set of the system of fuzzy equations which is constructed by the fuzzy equation as above,it also gives a sufficient and necessary condition that the solution set is nonempty,and it uses the additive generator to describe the greatest solution and all minimal solutions of the system.This paper shows us that every solution set has a minimal solution when the solution set is nonempty,and finally describes the solution set of the system.