Fuzzy axiom of choice, fuzzy Zorn's lemma and fuzzy Hausdorff maximal principle

Abstract

Equivalence relations and orderings are key concepts of mathematics. For those two relations, the formulation in fuzzy relations has been presented in the early days of fuzzy set theory. Utilizing the existing equivalence relations, we are unable to prove that fuzzy Zorn's lemma and fuzzy axiom of choice are equivalent. The main objective of this research is to redefine the fuzzy-order relation with some fundamental properties. Also, introduce fuzzy equivalence relation based on our developed fuzzy-order relation and proved that any equivalence relation is equivalent to fuzzy equivalence relation. Furthermore, introduce the fuzzy versions of the axiom of choice, Zorn's lemma, and well-ordering principle, such as the fuzzy axiom of choice, fuzzy Zorn's lemma, and fuzzy well-ordering principle and discuss the relations between them. We proposed a fuzzy version of the Hausdorff maximal principle. Moreover, we give the application of fuzzy Zorn's lemma in fuzzy ideals and fuzzy hausdroff maximal principle in fuzzy filters.