On the Structure and Properties of Hyperfuzzy and SuperhyperfuzzyIntervals

Abstract

A fuzzy set assigns to each element in a universe a membership degree in the interval [0, 1], effectively modeling imprecision and vagueness. A hyperfuzzy set extends this concept by mapping each element to a
nonempty subset of [0, 1], thereby representing multiple possible membership degrees and capturing both uncertainty and variability. An m, n-superhyperfuzzy set further generalizes this structure by assigning to
each nonempty element of the mth and the nth power-set hierarchy a nonempty subset of [0, 1], enabling the modeling of hierarchical and nested uncertainty. A fuzzy interval is a special kind of fuzzy set on R
that is both normal and convex, with each α-cut forming a nonempty closed interval, thus providing a precise framework for modeling uncertain numerical values. In this paper, we introduce and define the notions of Hyperfuzzy Interval and SuperHyperfuzzy Interval, and provide a brief investigation of their mathematical properties.