Order Relations and a Monotone Convergence Theorem in the Class of Fuzzy Sets on ℝ n

Abstract

Concerning with the topics of fuzzy decision processes, a brief survey on ordering of fuzzy numbers on ℝ is presented and an extension to that of fuzzy sets(numbers) on ℝ n are considered. This extension is a pseudo order ≼ K defined by a non-empty closed convex cone K and characterized by the projection into its dual cone K +. Especially a structure of the lattice is presented on the class of pyramid-type fuzzy sets. Moreover, we study the convergence of a sequence of fuzzy sets on ℝ n which is monotone w.r.t. the order ≼ K . Our study is carried out by restricting the class of fuzzy sets into the subclass in which the order ≼ K becomes a partial order so that a monotone convergence theorem is proved. This restricted subclass of fuzzy sets is created and characterized in the concept of a determining class. These results are applied to obtain the limit theorem for a sequence of fuzzy sets defined by the dynamic fuzzy system with a monotone fuzzy relation. Several figures are illustrated to comprehend our results.