On Principal Fuzzy Metric Spaces

Valentín Gregori (), Juan-José Miñana, Samuel Morillas and Almanzor Sapena
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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
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

Mathematics, 2022, vol. 10, issue 16, 1-10

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.

Keywords: fuzzy metric; Cauchy sequence; principal fuzzy metric; p -Cauchy sequence; completeness; completion (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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