This paper investigates function spaces of partially ordered sets with some directed family of projections. Given a fixed directed index set (I; ), we consider triples (D; ; (p i ) i2I ) consisting of a poset (D; ) and a monotone net (p i ) i2I of projections of D. We call them (I; )-indexed pop's (posets with projections). Our main purpose is to study structure preserving maps between (I; )-indexed pop's. Such a morphism respects both order and projections. In fact, we study weak homomorphisms as well as homomorphisms. In case of (I; ) = (N 0 ; ), weak homomorphisms are precisely all monotone maps that are non-expansive with regard to some canonical pseudo-ultrametric induced by the given sequence of projections. Weak homomorphisms become then homomorphisms if they are additionally compatible with a so-called weak weight function. Both weak homomorphisms and homomorphisms between two (I; )-indexed pop's turn out to induce (I; )-indexed pop's of their own. We prove that pr...

Posets with (I; )-indexed projections are triples (D; ; (p i ) i2I ). They consist of a partially ordered set (D; ) and a monotone net (p i ) i2I of projections on D with a fixed directed index set (I; ). In this paper we prove that there is a unique "completion" ( b D; b ; ( b p i ) i2I ) of (D; ; (p i ) i2I ). It is complete with respect to the uniformity induced by the kernels of all b p i . Moreover, it satisfies a universal property concerning the extension of special mappings ("homomorphisms"). If sup i2I p i = id D , then (D; ; (p i ) i2I ) can be viewed as a dense substructure of ( b D; b ; ( b p i ) i2I ). In the second part of this paper we obtain the ideal completion of (D; ) to be a poset with (I; )-indexed projections by extending the projections p i in a natural way. A comparison shows that the completion ( b D; b ; ( b p i ) i2I ) may be seen as a substructure of the ideal completion. We also investigate under which conditions these structures co...