Positive and negative parts in a lattice-ordered group ($g^{+}=g\vee 1,g^{-}=g^{-1}\vee 1$)

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.