The word idempotency signifies the study of semirings in which the addition operation is idempotent: a+ a = a. The best-known example is the max-plus

Results of Bunge and Funk and of Johnstone, providing constructively sound descriptions of the global points of the lower and upper powerlocales, are extended here to describe the generalized points and proved in a way that displays in a symmetric fashion two complementary treatments of frames: as suplattices and as preframes. We then also describe the points of the Vietoris powerlocale. In each of two special cases, an exponential $ D ($ being the Sierpinsky locale) is shown to be homeomorphic to a powerlocale: to the lower powerlocale when D is discrete, and to the upper powerlocale when D is compact regular. 1

The Kripke-Joyal semantics is used to interpret the fragment of intuitionistic logic containing ; ! and 8 in the category of locales. An axiomatic theory is developed that can be interpreted soundly in two ways, using either lower or upper powerlocales, so that pairs of separate results can be proved as single formal theorems. Openness and properness of maps between locales are characterized by descriptions using the logic, and it is proved that a locale is open iff its lower powerlocale has a greatest point. The entire account is constructive and holds for locales over any topos. 1

There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.

We introduce observation frames as an extension of ordinary frames. The aim is to give an abstract representation of a mapping from observable predicates to all predicates of a specific system. A full subcategory of the category of observation frames is shown to be dual to the category of T 0 topological spaces. The notions we use generalize those in the adjunction between frames and topological spaces in the sense that we generalize finite meets to infinite ones. We also give a predicate logic of observation frames with both infinite conjunctions and disjunctions, just like there is a geometric logic for (ordinary) frames with infinite disjunctions but only finite conjunctions. This theory is then applied to two situations: firstly to upper power spaces, and secondly we restrict the adjunction between the categories of topological spaces and of observation frames in order to obtain dualities for various subcategories of T 0 spaces. These involve non sober spaces. Contents 1 Introduct...

In concurrent constraint programming, divergence (i.e. an infinite computation) and failure are often identified. This is undesirable when modelling systems in which infinite behaviour arises naturally. This paper sets out a framework for an axiomatic and denotational view of concurrent constraint programming, and considers the relationship of both views as an instance of Stone duality. We propose a construction of a constraint system which allows both finite and infinite constraints. Subsequently, we provide semantic, topological definitions of safety, liveness and fairness properties in a given constraint system. The process language considered is parametrized by the constraint system. It allows the actions ask and tell, the prefix operator !, the (angelic) non-deterministic choice operator \Phi, the procedure call p(X), and the concurrency operator k. Keywords: concurrent constraint programming, liveness, fairness, semantic properties. This paper was partly written when the autho...

The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.

Stably compact spaces are a natural generalization of compact Hausdor spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain \closed relations" and study the resulting categorical properties. Apart from extending ordinary continuous maps, these morphisms have a number of pleasing properties, the most prominent, perhaps, being that they correspond to preframe homomorphisms on the localic side. We exploit this Stone-type duality to establish that the category of stably compact spaces and closed relations has bilimits.

The probabilistic power domain construction of Jones and Plotkin [6, 7] is defined by a construction on dcpo's. We present alternative definitions in terms of information systems `a la Vickers [12], and in terms of locales. On continuous domains, all three definitions coincide. 1 Introduction To model probabilistic and randomized algorithms in the semantic framework of dcpo's and Scott continuous functions, Jones and Plotkin introduce in [6, 7] the probabilistic power domain construction PD . It forms a computational monad in the sense of [8] in the category of dcpo's and continuous functions and various of its subcategories of `domains'. Every probabilistic powerdomain PDX is equipped with a family of binary operations + p indexed by a real number p between 0 and 1 such that A+ p B denotes the result of choosing A with probability p and B with probability 1 \Gamma p. Other applications of PD were found in [1]. The probabilistic powerdomain of the upper power space [10] of a second ...

Abstract valuations on a topological space X are functions that map open sets to 0, 1, or one value in between. We define a space of abstract valuations which for a continuous dcpo X is homeomorphic to the Plotkin power domain of X , and for a Hausdorff space X yields the Vietoris hyperspace of X . Thus we obtain a novel concrete representation of the Plotkin power domain. This representation is more similar to the standard representation of the probabilistic power domain than the previously known ones.