Ordering of Convex Fuzzy Sets : A Brief Survey and New Results

Abstract

AbstTuct Coneerning with the topics of a fuzzy max order, a briefsurvey on orderi-g of fuzzy numbers is presented im this article, and we wil] consider an extensien to that of fuzzy sets. An extension of the fuzzy max order as a pseudo order is investigated and defined on a class of fuzzy sets on R" (n) 1). This order is developed by using a non-empLy closed convex cone and characterized by the projection into its dual cone. Especially a structure of the lattice can be illustrated with the class of rectang]e-type fuzzy sets.

Fuzzy Distance and Fuzzy Norm

“…We define orders on F(R) based on orderings of level sets of fuzzy sets. 5,8,10,12]). Let a, b ∈ F (R).…”

“…The fuzzy max order for fuzzy numbers has been primarily defined in [12], and many researches have dealt with it. Then, the fuzzy max order for fuzzy numbers has been extended for fuzzy vectors in [10], for fuzzy sets which are closed, convex, normal, and support bounded in [8], and for general fuzzy sets in [5].…”

Fuzzy optimality relation for perceptive MDPs—the average case

“…Obviously, the binary relation satisfies the axioms of a partial order relation on F(R) (cf. [7,18]). 13 For r, s ∈ R, max{ r, s} and min{ r, s} are defined by max{ r, s}(y) := sup…”

“…[7]): (i) r s; (ii) r − s − and r + s + ( ∈ [0, 1]); (iii) max{ r, s} = s; (iv) min{ r, s} = r. Also we use the addition by 19…”

Perceptive Evaluation for the Optimal Discounted Reward in Markov Decision Processes

“…The binary relation on F(R) is defined as follows: For r, s ∈ F(R), r s if and only if (i) for any x ∈ R, there exists y ∈ R such that x ≤ y and r(x) ≤ s(y); (ii) for any y ∈ R, there exists x ∈ R such that x ≤ y and s(y) ≤ r(x): Obviously, the binary relation satisfies the axioms of a partial order relation on F(R) (cf. [6], [12]). …”