Abstract ordered compact convex sets and algebras of the (sub)probabilistic powerdomain monad over ordered compact spaces
The majority of categories used in denotational semantics are topological in nature. One of these is the category of stably compact spaces and continuous maps. Previously, Eilenberg-Moore algebras were studied for the extended probabilistic powerdomain monad over the category of ordered compact spaces X and order-preserving continuous maps in the sense of Nachbin. Relevant algebras were characterized as compact convex subsets of ordered locally convex topological vector spaces. In so doing, functional analytic tools were involved. The main accomplishments of this paper are as follows: the result mentioned is re-proved and is extended to the subprobabilistic case; topological methods are developed which defy an appeal to functional analysis; a more topological approach might be useful for the stably compact case; algebras of the (sub)probabilistic powerdomain monad inherit barycentric operations that satisfy the same equational laws as those in vector spaces. Also, it is shown that it is convenient, first, to embed these abstract convex sets in abstract cones, which are simpler to work with. Lastly, we state embedding theorems for abstract ordered locally compact cones and compact convex sets in ordered topological vector spaces.