A Study of n-ary Groupoids Based on Hesitant Fuzzy Sets

Abstract

Fuzzy set theory plays a vital role in solving many complicated problems dealing with uncertainty. An [Formula: see text]-ary algebraic system as a generalization of algebraic structures that allow for operations involving more than two elements. They provide a natural framework for representing and manipulating complex relationships and interactions among multiple elements, which is essential for solving many real-world problems. Many authors have studied fuzzy set theory over an [Formula: see text]-ary algebraic systems and have provided many fruitful results. Hesitant fuzzy set theory is a further extension of fuzzy set theory, which allows for the representation of uncertainty in decision making where decision makers may have multiple possible choices or may be uncertain about the degree of membership of an element in a given set. No authors have studied, by combining hesitant fuzzy set theory with an [Formula: see text]-ary algebraic systems. This motivates us to study hesitant fuzzy set over an [Formula: see text]-ary algebraic system. In this paper, we have applied hesitant fuzzy set theory in an [Formula: see text]-ary algebraic systems and introduced the notions of hesitant fuzzy subgroupoid. We provide the characterization of hesitant fuzzy [Formula: see text]-ary subgroupoid over an [Formula: see text]-ary groupoids and have studied their related properties. As an application of hesitant fuzzy set over an [Formula: see text]-ary groupoids, we have developed the concept of normal hesitant fuzzy subgroupoids over an [Formula: see text]-ary groupoids and have studied their various properties.