In this paper, we present a new way of defining the property of being WRP-Noetherian by making use of principal right poideals. Additionally, we provide a characterization of WRP-Noetherian ordered semigroups through their S-posets. Furthermore, we investigate how the property of being WRP-Noetherian behaves under some semigroup-theoretic constructions, like sub ordered semigroups, and quotients. Specifically, we establish necessary and sufficient conditions for the direct product of two ordered semigroups to be WRP-Noetherian.
