Ordered group

Remarks

Example: $\Z^2$ under addition, with $(a, b) \le (c, d)$ when $a \le c$ and $b \le d$, is an ordered group. If $a \le c$ and $b \le d$, then $a + x \le c + x$ and $b + y \le d + y$ for each $(x, y) \in \Z^2$. Non-example: on $\Z$, let $a \preceq b$ when $a = b$ or $(a, b) = (0, 1)$. This is a partial order, but $0 \preceq 1$ while $1 \not\preceq 2$, so adding $1$ breaks the condition. The positive cone of $G$ is $P = \set{g \in G : e \le g}$. It satisfies $PP \subseteq P$, $P \cap P^{-1} = \set{e}$ and $gPg^{-1} = P$ for each $g \in G$. Conversely, each subset $P$ with these three properties defines an ordered group by $a \le b \iff a^{-1} b \in P$. For an abelian group the condition is written additively: $a \le b$ implies $a + c \le b + c$.