# Order Relations and a Monotone Convergence Theorem in the Class of Fuzzy Sets on ℝn

• Masami Kurano
• Masami Yasuda
• Jun-ichi Nakagami
• Yuji Yoshida
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 73)

## 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.

## Keywords

Partial Order Fuzzy Number Binary Relation Closed Convex Cone Dual Cone
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 2001

## Authors and Affiliations

• Masami Kurano
• 1
• Masami Yasuda
• 1
• Jun-ichi Nakagami
• 1
• Yuji Yoshida
• 2
1. 1.Chiba UniversityChibaJapan
2. 2.Kitakyushu UniversityKitakyushuJapan