On the Theory of Hesitant Fuzzy Functions and Their Reduction to Fuzzy and Intuitionistic Fuzzy Function Models

Abstract

This paper develops a formal theoretical framework for hesitant fuzzy functions (HFFs) within the setting of hesitant fuzzy relations. Although hesitant fuzzy relations have been introduced in the literature, a corresponding function-based framework has not yet been established in a clear and systematic way. To address this gap, HFFs are defined as a special class of hesitant fuzzy relations satisfying suitable conditions that preserve consistency in hesitant fuzzy mappings. Within this framework, an HFF from  to  is interpreted as a mapping between hesitant fuzzy sets on  and , and several fundamental properties are studied, including equality, injectivity, surjectivity, composition, invertibility, and image and preimage behavior. Illustrative examples and counterexamples are also provided to clarify these notions in the hesitant fuzzy setting. In addition, several basic theorems are established to support the structural behavior of HFFs. The paper further shows that the proposed framework admits natural reductions to fuzzy functions (FFs) and intuitionistic fuzzy functions (IFFs) under suitable structural constraints on the co-membership functions of HFFs. This reduction highlights the unifying role of HFFs as a general framework for representing hesitation in functional settings and provides a basis for further developments in hesitant fuzzy mappings and related structures. This work not only deepens the understanding of hesitant fuzzy mappings but also bridges the gap between different fuzzy function models, with potential applications in decision-making, pattern recognition, and artificial intelligence systems dealing with vague or imprecise information.