Function Spaces of Posets with Projections

摘要

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,≤), we consider triples (D,≤,(Pi)_(i ∈ I)) with (D,≤) a poset and (Pi)_(i∈I) a monotone net of projections of D. We call then (I,≤)-pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,≤)-pop's Such 'homomorphisms' respect both order and projections. Any (I,≤)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,≤)-pop's turns out to form an (I,≤)-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,≤)-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∞-construction we get (I,≤)-pop's that are isomorphic to their own exponent. This yields new models for the untyped λ-calculus.