3 Cpo.v: Speci cation and properties of a cpo 7 3.1 Ordered type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.1 Associated equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.2 Setoid relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1.3 Dual order . ...

A non-empty subset of a topological space is irreducible if whenever it is covered by the union of two closed sets, then already it is covered by one of them. Irreducible sets occur in proliferation: (1) every singleton set is irreducible, (2) directed subsets (which of fundamental status in domain theory) of a poset are exactly its Alexandroff irreducible sets, (3) directed subsets (with respe...

We construct a complete lattice Z such that the binary supremum function sup : Z × Z → Z is discontinuous with respect to the product topology on Z × Z of the Scott topologies on each copy of Z. In addition, we show that bounded completeness of a complete lattice Z is in general not inherited by the dcpo C(X,Z) of continuous functions from X to Z where X may be any topological space and where o...

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 generalize the construction of the formal ball model for metric spaces due to A. Edalat and R. Heckmann to obtain computational models for separated Q-categories. We fully describe (a) Yoneda complete and (b) continuous Yoneda complete Q-categories via their formal ball models. Our results yield solutions to two open problems in the theory of quasi-metric spaces: we show that (a) a quasi-met...

If a strong monad M is used to deene the denotational semantics of a functional language with computations, a product operation ? : MX MY ! M(X Y) is needed to deene the semantics of pairing. Every strong monad is equipped with two standard products, which correspond to left-to-right and right-to-left evaluation. We study the algebraic properties of these standard products in general. Then we d...

valuations on a topological space X are functions that map open sets to 0, 1, or one value in between. We deene a space of abstract valuations which for a continuous dcpo X is homeomorphic to the Plotkin power domain of X, and for a Hausdorr space X yields the Vietoris hyperspace of X. Thus we obtain a novel concrete representation of the Plotkin power domain. This representation is more simila...

We give a constructive localic account of the completion of quasimetric spaces. In the context of Lawvere’s approach, using enriched categories, the points of the completion are flat left modules over the quasimetric space. The completion is a triquotient surjective image of a space of Cauchy sequences and can also be embedded in a continuous dcpo, the “ball domain”. Various examples and constr...

We prove that Keimel and Lawson's K-completion Kc of the simple valuation monad Vs defines a Kc∘Vs on each -category K. also characterise Eilenberg-Moore algebras as weakly locally convex K-cones, its algebra morphisms continuous linear maps. In addition, we explicitly describe distributive law over Kc, which allows us to show any (resp., convex, linear) topological cone is K-cone. give an exam...