Linearly ordered group

Remarks

Example: $\Z^2$ under addition with the lexicographic order is a linearly ordered group. If $(a, b) \le (c, d)$, then either $a < c$, or $a = c$ and $b \le d$, and both cases survive adding $(x, y)$ to both sides. Non-example: $\Z^2$ with the componentwise order is an ordered group, but $(1, 0)$ and $(0, 1)$ are not comparable, so the order is not total. In terms of the positive cone $P = \set{g : e \le g}$, the order is total if and only if $G = P \cup P^{-1}$. A linearly ordered group has no elements of finite order other than $e$: if $g > e$, then $g^n > e$ for all $n \ge 1$.