Abstract
This paper proposes a measurement theoretic framework in which the inter-relationship between inexact concepts and the fuzzy set theories is investigated. Each fuzzy set theory is a formal relational structure on the set of membership functions and regarded to represent the empirical relational structure of inexact concepts homomorphically. As the representation is uniquely characterized by permissible transformations on the set of membership functions, we construct some of them by using two types of mappings to show the sufficient conditions of the uniqueness. They are the endomorphisms on the universe of discourse where inexact concepts are defined, and on the image space of membership functions.
The discussion is also done about the scale of membership functional representation. Thus the capabilities and limitations of membership function in representing an inexact concept are pointed out explicitly.
