Year: 2025
Type: preprint
Abstract: Handling uncertainty in complex systems has led to a rich family of extensions to classical set theory, including fuzzy sets [1-3], intuitionistic fuzzy sets [4], and related models. A fuzzy set on a universe π assigns each element a degree of membership in [0, 1], capturing gradual inclusion. Multisets generalize ordinary sets by allowing repeated elements, and fuzzy multisets further permit each element to carry multiple membership values in (0, 1]. Hyperfuzzy sets extend this framework by mapping each element to a nonempty subset of [0, 1], thereby modeling both uncertainty and internal variability [5-7]. Building on these ideas, (π, π)superhyperfuzzy sets assign to each nonempty member of the πth and πth iterated powersets a nonempty family of subsets of [0, 1], enabling nested and hierarchical imprecision [8]. Although fuzzy and hyperfuzzy constructs have found applications in artificial intelligence, decision-making, algebra, topology, and control theory, more expressive structures-such as hyperfuzzy multisets and (π, π)superhyperfuzzy multisets-remain underexplored. In this paper, we introduce these novel constructs, present rigorous formal definitions, and investigate their fundamental properties. We anticipate that future work will develop computational methods and application-driven studies to fully leverage these advanced multiset frameworks.
Author: Takaaki Fujita
Language: English
Topic: Fuzzy and Soft Set Theory
Field: Decision Sciences
Domain: Social Sciences
Open Access status: gold
OpenAlex ID: W4413273679