An extension framework for creating operators and functions for intuitionistic fuzzy sets

Intuitionistic fuzzy sets, introduced by Atanassov in the mid-1980′s, represent a key extension of Zadeh's fuzzy sets. Zadeh's extension principle, a fundamental concept in fuzzy set theory, has been utilized to extend functions from the classical mathematics setting to the fuzzy setting. In this article, we propose an extension framework for intuitionistic fuzzy sets. The proposed approach offers several advantages: (i) Since an intuitionistic fuzzy function so-obtained is constructed from a bottom-up concrete manner, the behavior of the function, when applied to intuitionistic fuzzy sets, can be readily understood in terms of the behavior of its fuzzy counterpart. (ii) Certain mathematical properties fulfilled by the fuzzy counterpart in the fuzzy setting can often be translated into a corresponding statement that the analogous properties in the intuitionistic fuzzy setting are also fulfilled by the intuitionistic fuzzy function. (iii) The application of the extension framework to construct the intuitionistic fuzzy intersection and union operators has led us to the discovery of a natural spectrum of intuitionistic fuzzy intersection and union operators that have not been postulated in the literature before. Due to these advantages, the extension framework has the potential to facilitate greatly the theoretical and application developments of intuitionistic fuzzy sets. © 2024 Elsevier Inc.