Some Errors on Hesitant Fuzzy Set Theory

Hesitant fuzzy set theory serves as a valuable framework that has been extensively applied across various domains, including decision-making, attribute reduction, linguistic perception, among others. Hesitant fuzzy elements are discrete arrays, and the intersection and union operations for hesitant fuzzy sets differ from those defined for fuzzy sets. Consequently, certain erroneous propositions have emerged in the literature on hesitant fuzzy sets. This review examines some incorrect propositions found in studies related to hesitant fuzzy topological spaces, hesitant fuzzy approximation spaces and hesitant fuzzy algebra, and provides corresponding counterexamples in each incorrect proposition. The advancement of a mathematical knowledge system must be free from errors, as inaccuracies can compromise the integrity of the theoretical framework. It is essential that researchers rigorously scrutinize the flawed propositions identified in this work when further investigating hesitant fuzzy sets and their mathematical structures, thereby promoting the robust development of hesitant fuzzy set theory.