On Graev type ultra-metrics

TL;DR

This paper investigates Graev ultra-metrics, demonstrating their maximal property and showing their equivalence to certain ultra-metrics in specific cases, thereby advancing understanding of non-archimedean topological groups.

Contribution

It establishes the maximal property of Graev ultra-metrics and proves their equivalence to Savchenko and Zarichnyi's ultra-metric under certain conditions.

Findings

Graev ultra-metric has a maximal property.

The Graev ultra-metric coincides with Savchenko and Zarichnyi's ultra-metric when diameter ≤ 1.

The free non-archimedean balanced topological group is metrizable by a Graev ultra-metric.

Abstract

We study Graev ultra-metrics which were introduced by Gao. We show that the free non-archimedean balanced topological group defined over an ultra-metric space is metrizable by a Graev ultra-metric. We prove that the Graev ultra-metric has a maximal property. Using this property, among others, we show that the Graev ultra-metric associated with an ultra-metric space $(X,d)$ with diameter$\leq 1$ coincides with the ultra-metric $\hat{d}$ of Savchenko and Zarichnyi.