QRB, QFS, and the probabilistic powerdomain

AbstractWe 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 QFS-spaces, 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 proof that the probabilistic powerdomain of a QRB-domain is a QRB-domain. This improves upon a previous, similar result, which was limited to pointed, second-countable QRB-domains

Meet-continuity and locally compact sober dcpos

In this thesis, we investigate meet-continuity over dcpos. We give different equivalent descriptions of meet-continuous dcpos, among which an important characterisation is given via forbidden substructures. By checking the function space of such substructures we prove, as a central contribution, that any dcpo with a core-compact function space must be meet-continuous. As an application , this result entails that any cartesian closed full subcategory of quasicontinuous domains consists of continuous domains entirely. That is to say , both the category of continuous domains and that of quasicontinuous domains share the same cartesian closed full subcategories. Our new characterisation of meet-continuous dcpos also allows us to say more about full subcategories of locally compact sober dcpos which are generalisations of quasicontinuous domains. After developing some theory of characterising coherence and bicompleteness of dcpos, we conclude that any cartesian closed full subcategory of pointed locally compact sober dcpos is entirely contained in the category of stably compact dcpos or that of L-dcpos. As a by-product, our study of coherence of dcpos enables us to characterise Lawson-compactness over arbitrary dcpos

The Equivalence of QRB, QFS, and Compactness for Quasicontinuous Domains

In this article we show the equivalence of QRB, QFS, and compact quasicontinuous domains. QRB and QFS domains are generalizations of RB and FS domains to the setting of quasicontinuous domains and compactness means compactness in the Lawson topology. This equivalence extends in the algebraic setting to a quasicontinuous version of bifinite domains. The Smyth powerdomain is a basic tool in the proofs, and it is shown that quasicontinuous properties at the dcpo level typically have the corresponding continuous domain properties at the Smyth powerdomain level