Absolute value in a lattice-ordered group ($|g|=g\vee g^{-1}$)

The displayed join is nonnegative in a lattice-ordered group. In an additive Abelian group it becomes $|g|=g\vee (-g)$. This order-theoretic absolute value need not be a real scalar.

The two positive parts satisfy $g=g^{+}(g^{-}{)}^{-1}$, commute with one another, and have meet one. They isolate positive and negative displacement in the pointwise model of a lattice-ordered permutation group. These are group elements, unlike the scalar positive part of a real number.