 Ordered basis of a complete p-valued group

An ordered basis $(g_1,\dots ,g_d)$ of a complete finite-rank p-valued group gives each element a unique convergent expression $g_1^{{\lambda }_1}\cdots g_d^{{\lambda }_d}$ with ${\lambda }_i\in {\mathbb{Z}}_p$, and its valuation is the minimum of $\omega (g_i)+v_p({\lambda }_i)$ over the nonzero coordinates.