On various algebraic structures in multidimensional fuzzy sets

Abstract

One recent generalization of Zadeh’s fuzzy sets is the concept of multidimensional fuzzy sets. This work extends this concept further by introducing a generalized form of multidimensional fuzzy algebra. By focusing on multidimensional t-norms and t-conorms, we develop a comprehensive theory. This includes the notion of strong multidimensional fuzzy algebras and explores their properties, such as multidimensional groupoids, monoids, and groups. Additionally, we introduce equivalence relations on the collection of all multidimensional fuzzy sets and present an example of specific monoids within this collection. Finally, we demonstrate how a group structure can be imposed on a subcollection of orderless multidimensional fuzzy sets.