p-valuation

For a prime number $p$, a p-valuation is a separated filtration on a group such that every $x\ne 1$ satisfiesA group equipped with one is called a 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.

A p-valued group is p-saturated when it is complete and every $x$ withhas a pth root in the group.