On phi-extensions and fuzzy number functions

Abstract

This work studies the generalizations of Zadeh’s Extension Principle through the change of the minimum function by another aggregation function. The mathematical details involved are studied in order to assure that real number functions are extended to functions of fuzzy numbers whose image only contains fuzzy numbers. The notion of φ-extension was introduced in order to formalize the definition of these extended mappings in a way to avoid inconsistencies with the definition of fuzzy number used. Some algebraic properties are explored aiming to understand how some characteristics of a real number function are preserved. Some applications of φ-extensions are also studied, like the extensions of binary operations of real numbers. Particularly, a new definition of the product operation of fuzzy numbers is used to define a generalization of ordered weighted averaging functions for fuzzy numbers with fuzzy numbers as weights.