: We provide the rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) together with many examples and applications. Galois connections occur in profusion and are well-known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear; and thus, the whole situation typically becomes much easier to understand. KEY WORDS: Galois connection, closure operation, interior operation, polarity, axiality CLASSIFICATION: Primary: 06A15, 06--01, 06A06 Secondary: 54-01, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...

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We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic generalized in this fashion.

In this paper, we survey the use of order-theoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of order-theoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...

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.

Continuous maps of compact regular locales form a coreflective subcategory of the category of perfect maps of stably compact locales. The coreflection of a stably compact locale is given by the frame of Scott continuous nuclei and is referred to as its patch. We show that the patch of a stably compact locale is isomorphic to the patch of its Lawson dual.

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The paper provides conditions under which a given family H of sets is continuous with its Lawson dual b H isomorphic to another collection T of sets.

and such that the event f8s 62 H : ffl s = 1g is measurable. The correspondence ffl s = sup H6 3s /H /H = inf s62H ffl s ffl s = 1 , 9 H 6 3 s : /H = 1 Some drawbacks One would like to be able to pass from a multiple test ffi to an equivalent multiple cotest / and vice versa Is it true that S n T ` H; ffi T = 1 ) /H = 1 ? and that H " T = ;; /H = 1 ) ffi T = 1 ? One would like to relate the two strong coherency conditions ffi T = 0 ) 9s 2 T 8T to each other One would like to have the equality ffi T =1 S n T = in order to unambiguously dene ' Consider also the two requirements ffl s = 1 ) 9 T 3 s 8s 0 2 T : ffl s 0 = 1 ffl s = 1 ) 9 H 6 3 s 8s 0 62 H : ffl s 0 = 1 needed to associate ffl with a test and a cotest, resp Continuous posets Let L be a partially ordered set (poset) Assume that L is closed for ltering inma Write x AE y, if, whenever y ff z ff and fz ff g ff is ltering, then x z ff for some z ff L is called continuous, if fy 2 L : y AE xg is

A prime algebraic lattice can be characterised as isomorphic to the downwards-closed subsets, ordered by inclusion, of its complete primes. It is easily seen that the downwards-closed subsets of a partial order form a completely distributive algebraic lattice when ordered by inclusion. The converse also holds; any completely distributive algebraic lattice is isomorphic to such a set of downwards-closed subsets of a partial order. The partial order can be recovered from the lattice as the order of the lattice restricted to its complete primes. Consequently prime algebraic lattices are precisely the completely distributive algebraic lattices. The result extends to Scott domains. Several consequences are explored briefly: the representation of Berry’s dIdomains by event structures; a simplified form of information systems for completely distributive Scott domains; and a simple domain theory for concurrency.