Exercise 19, § 1

Let $G$ be an additive group and $(P_\alpha)$ a nonempty family of subsets of $G$ such that $P_\alpha + P_\beta \subset P_\alpha$ and $P_\alpha \cap (-P_\alpha) = \{0\}$; let $G_+$ denote the ordered group obtained from $G$ by taking $P_+$ as the set of positive elements. Show that $P + P \subset P$ and $P \cap (-P) = \{0\}$, where $P = \cap P_\alpha$; if $H$ is the ordered group obtained from $G$ by taking $P$ as the set of positive elements, show that $H$ is isomorphic to the diagonal subgroup of the product of the $G_\alpha$.