Lattice-ordered group

Remarks

Example: $\Z^2$ under addition with the componentwise order is an $\ell$-group. Here $(a, b) \vee (c, d) = (\max(a, c), \max(b, d))$ and $(a, b) \wedge (c, d) = (\min(a, c), \min(b, d))$. Non-example: on $\Z^2$, let $(a, b) \le (c, d)$ when $a \le c$ and $b = d$. This makes $\Z^2$ an ordered group, but $(0, 0)$ and $(0, 1)$ have no common upper bound, so it is not a lattice. In an $\ell$-group, multiplication distributes over joins and meets: $c(a \vee b) = ca \vee cb$ and $(a \vee b)c = ac \vee bc$. The lattice of an $\ell$-group is distributive. Every $\ell$-group is a torsion-free group. For $a \in G$ one writes $a^+ = a \vee e$, $a^- = a^{-1} \vee e$ and $\abs{a} = a \vee a^{-1}$, and then $a = a^+ (a^-)^{-1}$.