A Type of Fuzzy Metric and Its Applications

In this paper, we aim to investigate a type of lattice-valued fuzzy metric within the framework of L-topology. Firstly, we present a comprehensive construction theorem for this type of metric, utilizing the concept of L-quasi metric. Secondly, we provide an equivalent characterization through the use of C-nbd clusters, which are formed from all $B_r$: one of four types of basic spheres defined herein. Thirdly, recognizing that these four types of basic spheres serve as essential tools for characterizing various metrics, we meticulously examine the relationships among them and outline a series of topological properties associated with these metrics, which include their opening and closing characteristics, symmetrical property, and more. Finally, in addressing the corresponding symmetry problem between two types of basic spheres, namely $B_r\left(a\right)$ and $Q_r\left(a\right)$, we introduce a novel fuzzy p-metric and demonstrate tht the L-real line $\mathbf{R}\left(L\right)$ satisfies this fuzzy p-metric.