Lemma 2.1.8 of Glass' Partially Ordered Groups states:
$G$ is a directed group if and only if $\langle G^+\rangle=G$ (where $G^+=\{x:x\geq1\}$)
This doesn't make any sense to me. For example, $(\mathbb Z,+)$ is a directed group (every two elements have an upper and lower bound) so the lemma implies that you can add some number of positive integers together to create a negative integer, which clearly isn't true.
Is he using some weird definition of $\langle \cdot \rangle$ (e.g. where the inverses of elements are included)? What am I not understanding?
Given a group $G$ and a subset $S$ of $G$, $\langle S\rangle$ is (typically) defined to be the smallest subgroup of $G$ containing $S$ (i.e.: the intersection of all subgroups of $G$ containing $S$). Thus, inverses are included.