The mathematical framework of Stone duality is used to synthesize a number of hitherto separate developments in Theoretical Computer Science: • Domain Theory, the mathematical theory of computation introduced by Scott as a foundation for denotational semantics. • The theory of concurrency and systems behaviour developed by Milner, Hennessy et al. based on operational semantics. • Logics of programs. Stone duality provides a junction between semantics (spaces of points = denotations of computational processes) and logics (lattices of properties of processes). Moreover, the underlying logic is geometric, which can be computationally interpreted as the logic of observable properties—i.e. properties which can be determined to hold of a process on the basis of a finite amount of information about its execution. These ideas lead to the following programme:

The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps,

this paper are algebraic dcpos, and many of the points discussed here will be needed later in the special case. 2 They provide a simple example to illustrate the "Display categories" in Section 3.2

. To any entailment relation [Sco74] we associate a distributive lattice. We use this to give a construction of the product of lattices over an arbitrary index set, of the Vietoris construction, of the embedding of a distributive lattice in a boolean algebra, and to give a logical description of some spaces associated to mathematical structures. 1 Introduction Most spaces associated to mathematical structures: spectrum of a ring, space of valuations of a field, space of bounded linear functionals, . . . can be represented as distributive lattices. The key to have a natural definition in these cases is to use the notion of entailment relation due to Dana Scott. This note explains the connection between entailment relations and distributive lattices. An entailment relation may be seen as a logical description of a distributive lattice. Furthermore, most operations on distributive lattices are simpler when formulated as operations on entailment relations. A special kind of distribu...

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

Abstract. In this paper we discuss a uniform method for constructing free modal and distributive modal algebras. This method draws on works by (Abramsky 2005) and (Ghilardi 1995). We revisit the theory of normal forms for modal logic and derive a normal form representation for positive modal logic. We also show that every finitely generated free modal and distributive modal algebra axiomatised by equations of rank 1 is a reduct of a temporal algebra. 1

A nucleus on a frame is a finite-meet preserving closure operator. The nuclei on a frame form themselves a frame, with the Scott continuous nuclei as a subframe. We refer to this subframe as the patch frame. We show that the patch construction exhibits the category of compact regular locales and continuous maps as a coreflective subcategory of the category of stably compact locales and perfect maps, and the category of Stone locales and continuous maps as a coreflective subcategory of the category of spectral locales and spectral maps. We relate our patch construction to Banaschewski and Brummer's construction of the dual equivalence of the category of stably compact locales and perfect maps with the category of compact regular biframes and biframe homomorphisms. Keywords: Frame of nuclei, Scott continuous nuclei, patch topology, stably locally compact locales, perfect maps, compact regular locales. AMS Classification: 06A15, 06B35, 06D20, 06E15, 54C10, 54D45, 54F05. 1 Introduction ...

ASD (Abstract Stone Duality) is a re-axiomatisation of general topology in which the topology on a space is treated, not as an infinitary lattice, but as an exponential object of the same category as the original space, with an associated lambda-calculus. In this paper, this is shown to be equivalent to a notion of computable basis for locally compact sober spaces or locales, involving a family of open subspaces and accompanying family of compact ones. This generalises Smyth’s effectively given domains and Jung’s Strong proximity lattices. Part of the data for a basis is the inclusion relation of compact subspaces within open ones, which is formulated in locale theory as the way-below relation on a continuous lattice. The finitary properties of this relation are characterised here, including the Wilker condition for the cover of a compact space by two open ones. The real line is used as a running example, being closely related to Scott’s domain of intervals. ASD does not use the category of sets, but the full subcategory of overt discrete objects plays this role; it is an arithmetic universe (pretopos with lists). In particular, we use this subcategory to translate computable bases for classical spaces into objects in the ASD calculus.

Abstract. Joins in the frame of nuclei are hard to describe explicitly because a pointwise join of a set of closure operators on a complete lattice fails to be idempotent in general. We calculate joins of nuclei as least fixed points of inflationary operators on prenuclei. Using a recent fixed-point theorem due to Pataraia, we deduce an induction principle for joins of nuclei. As an illustration of the technique, we offer a simple (and also intuitionistic) proof of the localic Hofmann–Mislove Theorem. 1.