There are three common variants: the Plotkin, upper, and lower power domains.
This flexibility is genuine; for example, in some concurrent systems it is natural to impose the condition that every message sent must eventually be delivered.
The abstract characterisation of powerdomains is often the easiest way to work with them, because explicit descriptions are so intricate.
Models of the lower powertheory are called inflationary semilattices; there is an additional requirement that the operator behave a little like a join for the order.
For the upper powertheory, models are called deflationary semilattices; here, the operator behaves a little like a meet.
The lower power domain can be defined by In other words, P[D] is the collection of downward-closed subsets of D that are also closed under existing least upper bounds of directed sets in D. Note that while the ordering on P[D] is given by the subset relation, least upper bounds do not in general coincide with unions.
Hewitt [2006] constructed a power domain for the Actor model (which is technically simpler and easier to understand than Clinger's model) building on a base domain of timed Actor event diagrams, which is complete.
The idea is to attach an arrival time for each message received by an Actor.