Wolfenstein inequality for normal-valued lattice-ordered groups ($|f||g|\le |g{|}^2|f{|}^2$)

A lattice-ordered group is normal-valued precisely when the displayed inequality holds for all its elements. The theorem is proved without using it as a definition. In the forward direction a normal value has a linearly ordered quotient of its cover with no proper nontrivial convex subgroup. That quotient is Archimedean, hence Abelian by Archimedean linearly ordered groups are Abelian. Apply squared bound from comparison at values to $p=b,q=a,x=ab$. For the reverse direction, power cofinality in the cover of a value excludes a conjugate $u=a^{-1}va$ outside the value: its right cosets $V{u}^n$ are all bounded by $Va$, contradicting cofinality above $V{a}^2$. Left-coset cofinality treats $av{a}^{-1}$. Every element is a quotient of nonnegative lattice parts, proving normality in the cover. In a linearly ordered group, comparison of $a,b\ge 1$ immediately gives $ab\le \max (a,b{)}^2\le b^2{a}^2$, so every such group is normal-valued.