  A regular ordered semigroup $S$ is called right inverse if every principal
left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent.
Here we explore the theory of right inverse ordered semigroups. We show that a
regular ordered semigroup is right inverse if and only if any two right
inverses of an element $a\in S$ are $\mathcal{R}$-related. Furthermore,
different characterizations of right Clifford, right group-like, group like
ordered semigroups are done by right inverse ordered semigroups. Thus a
foundation of right inverse semigroups has been developed.
