Principal convex lattice subgroup ($⟨u{⟩}_c=\{g:u^{-n}\le g\le u^n\text{ for some }n\}$)

For $u\ge 1$ in a lattice-ordered group, the displayed set is the smallest convex lattice subgroup containing $u$. Products of two such bounds are bounded by a larger power of $u$; inversion and the lattice operations preserve the bounds. This gives an explicit test for membership.