Posets with Projections and their Morphisms

This paper investigates function spaces of partially ordered sets with some directed family of projections. Given a xed 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 properties such as completeness and compactness of the canonical uniformity induced by the projections are inherited by the function spaces. Moreover, we obtain several cartesian closed categories of (I;)-indexed pop's.