On graev type ultra-metrics

We study Graev ultra-metrics which were introduced by Gao [3]. 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≤ 1 coincides with the ultra-metric d of Savchenko and Zarichnyi [12].