Poset-enriched categories and free exact completions

We give an elementary construction of the exact completion of a weakly lex category for categories enriched in the cartesian closed category $\mathsf{P}\mathsf{o}\mathsf{s}$ of partially ordered sets. Paralleling the ordinary case, we characterize categories which arise as such completions in terms of projective objects. We then apply the results to categories of Eilenberg-Moore algebras for monads on $\mathsf{P}\mathsf{o}\mathsf{s}$. In particular, we show that every variety of ordered algebras is the exact completion of a subcategory on certain free algebras, thereby answering a question of A. Kurz and J. Velebil.