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Fuzzy Metrics in Terms of Fuzzy Relations

Author

Listed:
  • Olga Grigorenko

    (Institute of Mathematics and CS, University of Latvia, LV-1459 Riga, Latvia)

  • Alexander Šostak

    (Institute of Mathematics and CS, University of Latvia, LV-1459 Riga, Latvia
    Department of Mathematics, University of Latvia, LV-1004 Riga, Latvia)

Abstract

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E - d -metric, which is a fuzzification of the expression d ( x , y ) = t . Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value d ( x , y ) and the other is a number within the range [ 0 , + ∞ ) .

Suggested Citation

  • Olga Grigorenko & Alexander Šostak, 2023. "Fuzzy Metrics in Terms of Fuzzy Relations," Mathematics, MDPI, vol. 11(16), pages 1-13, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3528-:d:1217876
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    References listed on IDEAS

    as
    1. Valentín Gregori & Juan-José Miñana & Samuel Morillas & Almanzor Sapena, 2022. "On Principal Fuzzy Metric Spaces," Mathematics, MDPI, vol. 10(16), pages 1-10, August.
    2. Salvador Romaguera & Pedro Tirado, 2020. "Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results," Mathematics, MDPI, vol. 8(2), pages 1-7, February.
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    Citations

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    Cited by:

    1. Valentín Gregori & Juan-José Miñana & Bernardino Roig & Almanzor Sapena, 2024. "On Completeness and Fixed Point Theorems in Fuzzy Metric Spaces," Mathematics, MDPI, vol. 12(2), pages 1-7, January.

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