Locally convex cones and the Schröder-Simpson theorem

Abstract

This paper paper has two goals: Firstly, to present the conceptual proof of the Schröder-Simpson theorem. The Schröder-Simpson theorem is stated in terms of domain theory and uses directed complete partially ordered cones and Scott-continuous maps. These structures are used to model probabilistic phenomena in denotational semantics. The proof presented here relies on another generalization of vector spaces to an asymmetric setting – cones with convex quasiuniform structures – which has not been used in the semantic community until now. The second goal of this paper is to introduce these two parallel developments of asymmetric generalizations of topological vector spaces in the style of a survey. There are the order theoretical and topological point of view on one hand, and the quasiuniform aspect on the other hand. Both developments had been pursued in parallel until now. The aim is to point out the close connection of these developments to the community working on asymmetric normed and asymmetric locally convex spaces.