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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… (More)

We investigate the notion of distance on domains. In particular, we show that measurement is a fundamental concept underlying partial metrics by proving that a domain in its Scott topology is partially metrizable only if it admits a measurement. Conversely, the natural notion of a distance associated with a measurement not only yields meaningful partial… (More)

Our work is a fundamental study of the notion of approximation in Q-categories and in (U, Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Q-and (U, Q)-categories. We fully characterize continuous Q-categories (resp. (U,… (More)

This paper is about a generalization of Scott's domain theory in such a way that its definitions and theorems become meaningful in quasimetric spaces. The generalization is achieved by a change of logic: the fundamental concepts of original domain theory (order, way-below relation , Scott-open sets, continuous maps, etc.) are interpreted as predicates that… (More)

We provide a method for checking if a given auxiliary relation on a poset is the approximation relation on a domain. 1 Motivation If (P, ⊑) is a partial order, then the approximation (called also the way-below relation) is defined for x, y ∈ P as x ≪ y if and only of for any directed subset D of P , if y ⊑ ↑ D, then x ⊑ d for some d ∈ D (↑ D denotes the… (More)

In this paper we study the interplay between metric and order completeness of semantic domains equipped with generalised distances. We prove that for bounded complete posets directed-completeness and partial metric completeness are interdefinable. Moreover, we demonstrate that Lawson-compact, countably based domains are precisely the compact pmetric spaces… (More)

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