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

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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.

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Acknowledgements

The authors are highly thankful to the Editor-in-chief and the referees for their valuable comments and suggestions for improving the quality of our paper.

Funding

This research is partially supported by a grant of the National Natural Science Foundation of China (Grant Number 11971384).

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Authors and Affiliations

  1. School of Science, Xi’an Polytechnic University, Xi’an, 710048, China

    Xiao Long Xin

  2. School of Mathematics, Northwest University, Xi’an, 710127, China

    Rana Muhammad Zulqarnain & Xiao Long Xin

  3. Department of Mathematics Education, Gyeongsang National University, Jinju, 52828, Korea

    Young Bai Jun

Authors
  1. Rana Muhammad Zulqarnain
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  2. Xiao Long Xin
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  3. Young Bai Jun
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The authors contributed to each part of this paper equally. The authors read and approved the final manuscript.

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Correspondence to Xiao Long Xin.

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Zulqarnain, R.M., Xin, X.L. & Jun, Y.B. Fuzzy axiom of choice, fuzzy Zorn’s lemma and fuzzy Hausdorff maximal principle. Soft Comput 25, 11421–11428 (2021). https://doi.org/10.1007/s00500-021-06000-z

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