Positive cone of an ordered group
= Positive cone of an ordered group
{title2=$G_{\ge1}=\{g:1\le g\}$}
The nonnegative cone determines the <partial order> by $a\le b\iff a^{-1}b\in G_{\ge1}$. It contains the <identity element>, is closed under products and <conjugation>, and intersects its inverse only in the identity. Conversely these conditions define a two-sided invariant <partial order>. Its strictly positive part omits the identity; conventions for $G^+$ differ, so specify which is intended.