 Power cofinality in the cover of a value

Under the inequality in one-sided bounds for adjoining a positive element, if $a\ge 1$ belongs to the cover of a value in a lattice-ordered group $V^{\ast }$ but not to $V$, then $⟨V,a{⟩}_c=V^{\ast }$. Thus every $b\ge 1$ in the cover is bounded above by both $v{a}^n$ and $a^{m}w$ for suitable nonnegative $v,w\in V$ and integers $m,n\ge 0$. The corresponding right and left coset bounds give the displayed cofinality, without first assuming that $V$ is normal. Left-coset order is obtained by inversion of right-coset order.