On Principal Fuzzy Metric Spaces

Abstract

[EN] 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.Samuel Morillas acknowledges financial support from Ministerio de Ciencia e Innovacion of Spain under grant PID2019-107790RB-C22 funded by MCIN/AEI/10.13039/501100011033. JuanJose Minana acknowledges financial support from Proyecto PGC2018-095709-B-C21 financiado por MCIN/AEI/10.13039/501100011033 y FEDER "Una manera de hacer Europa" and from project BUGWRIGHT2. This last project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 871260. Also acknowledge support of Generalitat Valenciana under grant CIAICO/2021/137. This publication reflects only the authors' views and the European Union is not liable for any use that may be made of the information contained therein.Gregori Gregori, V.; Miñana, J.; Morillas, S.; Sapena Piera, A. (2022). On Principal Fuzzy Metric Spaces. Mathematics. 10(16):1-10. https://doi.org/10.3390/math10162860110101