If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale (and also for its openness). Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of ...

Abstract We consider the binary supremum function $$\sup :Z\times Z\rightarrow Z$$ sup : Z × → on a sup semilattice Z and its topological properties with respect to Scott topology product topology. It is well known that...

In this paper, the definition of meet-continuity on $L$-directedcomplete posets (for short, $L$-dcpos) is introduced. As ageneralization of meet-continuity on crisp dcpos, meet-continuity on$L$-dcpos, based on the generalized Scott topology, ischaracterized. In particular, it is shown that every continuous$L$-dcpo is meet-continuous and $L$-continuous retracts ofmeet-continuous $L$-dcpos are al...

We consider a programming language that can manipulate both classical and quantum information. Our is type-safe designed for variational programming, which hybrid classical-quantum computational paradigm. The subsystem of the Probabilistic FixPoint Calculus (PFPC), lambda calculus with mixed-variance recursive types, term recursion probabilistic choice. first-order linear type system two subsys...

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...