From Fuzzy Sets to Interval-Valued and Atanassov Intuitionistic Fuzzy Sets: A Unified View of Different Axiomatic Measures

This paper examines a broad collection of axiomatic definitions from various and diverse contexts within the domain of fuzzy sets (FS) to evaluate their respective extensions to the case of interval-valued fuzzy (IVF) sets and intuitionistic fuzzy (IF) sets from a purely formal point of view. We conclude that a large number of such extensions follow similar formal procedures. This fact allows us to formulate a general procedure that encompasses all the reviewed extensions as particular cases of it. The new general formulation allows us to identify three different procedures to derive the corresponding extension to the field of IVF or IF sets from a specific real-valued measure in the context of FSs. These three processes agglutinate a multitude of particular constructions found in the literature.