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A Primer On Galois Connections
 York Academy of Science
, 1992
"... : 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 wellknown to most mathematicians who deal with order theory; they seem to be less known to to ..."
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: 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 wellknown 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, 0601, 06A06 Secondary: 5401, 54B99, 54H99, 68F05 0. INTRODUCTION Mathematicians are familiar with the following situation: there are two "worlds" and t...
Semantic Domains for Combining Probability and NonDeterminism
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2005
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Multi Lingual Sequent Calculus and Coherent Spaces
 Fundamenta Informaticae
, 1997
"... 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 ge ..."
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Cited by 16 (7 self)
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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.
Stably Compact Spaces and Closed Relations
, 2001
"... Stably compact spaces are a natural generalization of compact Hausdorff 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 ..."
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Cited by 13 (2 self)
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Stably compact spaces are a natural generalization of compact Hausdorff 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 Stonetype duality to establish that the category of stably compact spaces and closed relations has bilimits.
Topology, Domain Theory and Theoretical Computer Science
, 1997
"... In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from ..."
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Cited by 10 (2 self)
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In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic 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...
Bicontinuous Function Spaces
, 1999
"... Given a sober space (X; O(X)) and a complete lattice L in its Scotttopology, we study the function space [X ! L] of all continuous maps f : X ! L, ordered pointwise. We show that this partial order is a bicontinuous lattice (i.e. the lattice and its order dual are continuous) if and only if L is bi ..."
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Cited by 3 (1 self)
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Given a sober space (X; O(X)) and a complete lattice L in its Scotttopology, we study the function space [X ! L] of all continuous maps f : X ! L, ordered pointwise. We show that this partial order is a bicontinuous lattice (i.e. the lattice and its order dual are continuous) if and only if L is bicontinuous, X is a continuous domain and O(X) is its Scotttopology. This extends known results on the continuity of the space [X ! L]. The techniques are novel in the theory of continuous lattices in that they employ a representation of the dual of [X ! L] as the lattice of maps preserving all suprema of type ¯: O(X) ! L op , where L op is the order dual of L. We specialize these results down to two classes of bicontinuous lattices: linear FSlattices and completely distributive lattices. 1 Introduction For a complete lattice L, a subset U is Scott open if it is an upper set that is inaccessible by directed suprema: if D ` L is directed and W D 2 U , then D " U 6= ;: This defines a...
The patch frame of the Lawson dual of a stably continuous frame.Unpublished research note
 School of Computer Science, St Andrews University
, 2000
"... 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 compa ..."
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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.
Prime Algebraicity
, 2009
"... A prime algebraic lattice can be characterised as isomorphic to the downwardsclosed subsets, ordered by inclusion, of its complete primes. It is easily seen that the downwardsclosed subsets of a partial order form a completely distributive algebraic lattice when ordered by inclusion. The converse ..."
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A prime algebraic lattice can be characterised as isomorphic to the downwardsclosed subsets, ordered by inclusion, of its complete primes. It is easily seen that the downwardsclosed 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 downwardsclosed 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.
A HofmannMislove theorem for bitopological spaces
, 2008
"... We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that ar ..."
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We present a Stone duality for bitopological spaces in analogy to the duality between topological spaces and frames, and discuss the resulting notions of sobriety and spatiality. Under the additional assumption of regularity, we prove a characterisation theorem for subsets of a bisober space that are compact in one and closed in the other topology. This is in analogy to the celebrated HofmannMislove theorem for sober spaces. We link the characterisation to Taylor’s and Escardó’s reading of the HofmannMislove theorem as continuous quantification over a subspace.