Partial Functions, Ordered Categories, Limits and Cartesian Closure

Abstract
Author CBJ

Partial maps are naturally ordered according to their extent of definition. Constructions on partial maps should preserve this order so that as a component or module in a construction (such as pairing or composition) becomes more defined then so does the construct as a whole, without changing any of its existing values. Yet despite the vast literature devoted to partial maps, this principle of modularity has not been given systematic attention. To do so the partial maps must be viewed as the morphisms of an ordered category, and the theory of limits, etc. developed in this context.