Definition 4

— In a lattice ordered group $G$ the positive part (resp. negative part, absolute value) of an element $x \in G$ is the element $\sup(x, 0)$ (resp. $\sup(-x, 0)$, $\sup(x, -x)$), which is denoted $x^+$ (resp. $x^-$, $|x|$).

Despite its name, the negative part $x^-$ is a positive element.

Clearly $x^- = (-x)^+$ and $|-x| = |x|$. Let us also note the following formulae, the first of which is an immediate consequence of the definitions and of the invariance of the ordering under translation, and the second of which follows from the first by Prop. 7 of VI, p. 10: