Function spaces of posets with projections

This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections. More precisely, given a fixed directed index set ( I,less than or equal to), we consider triples (D,less than or equal to, (p(i)) (iis an element ofI)) with (D,less than or equal to) a poset and (p(i))(iis an element ofI) a monotone net of projections of D. We call them (I,less than or equal to)- pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,less than or equal to)- pop's. Such 'homomorphisms' respect both order and projections. Any (I,less than or equal to)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,<)- pop's turns out to forman (I,less than or equal to)-pop itself. We show that its uniformity is the uniformity of uniform convergence. This enables us to prove that properties such as completeness and compactness transfer to 'function pop's'. Concerning categorical properties of (I,less than or equal to)- pop's, we will see that we are in a lucky situation from a computer scientist's point of view: we obtain Cartesian closed categories. Moreover, by a D-infinity-construction we get (I,less than or equal to)-pop's that are isomorphic to their own exponent. This yields new models for the untyped lambda-calculus.