  An ordered semigroup $S$ is right $\pi$-inverse if it is $\pi$-inverse but
not conversely. So the question arises under what condition the converse holds.
In this paper we study nil-extensions of simple and right $\pi$-inverse ordered
semigroups and prove that $S$ is right $\pi$-inverse if and only if $S$ is
$\pi$-inverse in a $t$-Archimedean ordered semigroup. Moreover, we characterize
complete semilattice of nil-extensions of simple and right $\pi$-inverse
ordered semigroups.
