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On Principal Fuzzy Metric Spaces

Author

Listed:
  • Valentín Gregori

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/Paranimf, 1, 46730 Grao de Gandia, Spain)

  • Juan-José Miñana

    (Departament de Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, Carretera de Valldemossa km. 7.5, 07122 Palma, Spain
    Institut d’ Investigació Sanitària Illes Balears (IdISBa), Hospital Universitari Son Espases, 07120 Palma, Spain)

  • Samuel Morillas

    (Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camí de Vera, s/n, 46002 Valencia, Spain)

  • Almanzor Sapena

    (Instituto de Investigación para la Gestión Integrada de Zonas Costeras, Universitat Politècnica de València, C/Paranimf, 1, 46730 Grao de Gandia, Spain)

Abstract

In this paper, we deal with the notion of fuzzy metric space ( X , M , ∗ ) , or simply X , due to George and Veeramani. It is well known that such fuzzy metric spaces, in general, are not completable and also that there exist p -Cauchy sequences which are not Cauchy. We prove that if every p -Cauchy sequence in X is Cauchy, then X is principal, and we observe that the converse is false, in general. Hence, we introduce and study a stronger concept than principal, called strongly principal. Moreover, X is called weak p -complete if every p -Cauchy sequence is p -convergent. We prove that if X is strongly principal (or weak p -complete principal), then the family of p -Cauchy sequences agrees with the family of Cauchy sequences. Among other results related to completeness, we prove that every strongly principal fuzzy metric space where M is strong with respect to an integral (positive) t -norm ∗ admits completion.

Suggested Citation

  • Valentín Gregori & Juan-José Miñana & Samuel Morillas & Almanzor Sapena, 2022. "On Principal Fuzzy Metric Spaces," Mathematics, MDPI, vol. 10(16), pages 1-10, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:16:p:2860-:d:885427
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    Cited by:

    1. Olga Grigorenko & Alexander Šostak, 2022. "Fuzzy Extension of Crisp Metric by Means of Fuzzy Equivalence Relation," Mathematics, MDPI, vol. 10(24), pages 1-15, December.
    2. Olga Grigorenko & Alexander Šostak, 2023. "Fuzzy Metrics in Terms of Fuzzy Relations," Mathematics, MDPI, vol. 11(16), pages 1-13, August.

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