Purity in the category of S-posets

In this paper, purity of S-posets over a pomonoid S is investigated. We first study some basic properties of absolutely po-pure S-posets. Among other results, it is proved that every regular injective S-poset is absolutely po-pure, and every absolutely po-pure inequationally compact S-poset is regular injective. Then, using the notion of semi-finitely presented S-poset based on the finitely induced S-poset congruence, we find an equivalent condition for an S-poset to be 1-po-pure in a regular extension. Finally, a characterization of an absolutely 1-po-pure S-poset is presented.