In this paper, we introduce the notion of super finitely separating functions which gives a characterization of RB-domains. Then we prove that FS-domains and RB-domains are equivalent in some special cases by the following three claims: a dcpo is an RB-domain if and only if there exists an approximate identity for it consisting of super finitely separating functions; a consistent join-semilatti...

We show that the first author’s QRB-domains coincide with Li and Xu’s QFS-domains, and also with Lawson-compact quasi-continuous dcpos, with stably-compact locally finitary compact spaces, with sober QFSspaces, and with sober QRB-spaces. The first three coincidences were discovered independently by Lawson and Xi. The equivalence with sober QFS-spaces is then applied to give a novel, direct proo...

We present two characterisations of FS domains, using the upper and the lower power domain construction, respectively. These characterisations have a common structure and thus can be generalised from the two power constructions to arbitrary premonads. The resulting classes of domains are considered in general, and some special instances are studied in greater detail. They form classes of domain...

Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains diier in the nature of the underlying semiring logic and in logical constraints on the second order predicates.

A novel upper power domain construction is deened by means of strongly compact sets. Its power domains contain less elements than the classical ones in terms of compact sets, but still admit all necessary operations, i.e. they contain less junk. The notion of strong compactness allows a proof of stronger properties than compactness would, e.g. an intrinsic universal property of the upper power ...

The initial lower and upper power domain constructions P _ and P ^ commute under composition for all cpos. The common result P _ (P ^ X) and P ^ (P _ X) is the free frame over the cpo X.

Stably compact spaces Xs can be described as being derived from compact ordered spaces X by weakening their topology to the open upper sets. In this paper the probabilistic powerdomain of a stably compact space Xs is investigated using the compact ordered space X and classical tools from measure theory and functional analysis. This allows to derive in a unified and elegant way known and new res...