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20
About Stone's notion of Spectrum
 J. Pure Appl. Algebra
, 2000
"... Introduction Stone duality between Boolean algebra and compact totally disconnected spaces provides an algebraic and pointfree presentation of a large class of topological spaces. A generalisation of this presentation, also due to Stone, is described in [Johnstone, Stone]: the class of spaces is l ..."
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Cited by 23 (6 self)
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Introduction Stone duality between Boolean algebra and compact totally disconnected spaces provides an algebraic and pointfree presentation of a large class of topological spaces. A generalisation of this presentation, also due to Stone, is described in [Johnstone, Stone]: the class of spaces is larger, we get all compact Hausdorff spaces, and the representation is still algebraic, using divisible archimedian rings. These rings are now torsion free, contrary to the case of Boolean rings. Actually, these rings appear implicitely in analysing problems of measure on Boolean algebras [Tarski]. We give here a variation of the treatment of [Johnstone, Stone] 1 which can be seen also as a constructive "real" version of Gelfand duality in the style of [BM]. Our main result is a localic proof of the fact that the uniform norm of the Gelfand transform of an element is equal to its norm. 1 Preordered Ring We start from a ring A with a p
Entailment Relations and Distributive Lattices
, 1998
"... . 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 ..."
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Cited by 18 (4 self)
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. 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...
Étale groupoids and their quantales
, 2004
"... We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantal ..."
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Cited by 16 (7 self)
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We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal
A Pointfree approach to Constructive Analysis in Type Theory
, 1997
"... The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from ..."
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Cited by 7 (0 self)
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The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the HeineBorel covering theorem is proved constructively. The basic definitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the HahnBanach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional MartinLöf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...
The HahnBanach Theorem in Type Theory
, 1997
"... We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and HahnBanach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topol ..."
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Cited by 7 (0 self)
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We give the basic deønitions for pointfree functional analysis and present constructive proofs of the Alaoglu and HahnBanach theorems in the setting of formal topology. 1 Introduction We present the basic concepts and deønitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and HellyHahnBanach 1 theorems. Earlier pointfree formulations of the HahnBanach theorem, in a topostheoretic setting, were presented by Mulvey and Pelletier (1987,1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to deøne the objects under analysis as formal points of a suitable formal space. After this has...
Localic completion of generalized metric spaces I
, 2005
"... Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a complet ..."
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Cited by 4 (0 self)
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Abstract. Following Lawvere, a generalized metric space (gms) is a set X equipped with a metric map from X 2 to the interval of upper reals (approximated from above but not from below) from 0 to ∞ inclusive, and satisfying the zero selfdistance law and the triangle inequality. We describe a completion of gms’s by Cauchy filters of formal balls. In terms of Lawvere’s approach using categories enriched over [0, ∞], the Cauchy filters are equivalent to flat left modules. The completion generalizes the usual one for metric spaces. For quasimetrics it is equivalent to the Yoneda completion in its netwise form due to Künzi and Schellekens and thereby gives a new and explicit characterization of the points of the Yoneda completion. Nonexpansive functions between gms’s lift to continuous maps between the completions. Various examples and constructions are given, including finite products. The completion is easily adapted to produce a locale, and that part of the work is constructively valid. The exposition illustrates the use of geometric logic to enable pointbased reasoning for locales. 1.
Geometric HahnBanach theorem
 Math. Proc. Cambridge Philos. Soc
"... In [MP2] is proved in a constructive way the following result: a point x lies in a compact convex set K in a normed space if and only if it lies within any bound for K (a bound for K being intuitively a parallel pair of hyperplanes between which K lies). This is given as an application of the locali ..."
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Cited by 4 (2 self)
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In [MP2] is proved in a constructive way the following result: a point x lies in a compact convex set K in a normed space if and only if it lies within any bound for K (a bound for K being intuitively a parallel pair of hyperplanes between which K lies). This is given as an application of the localic form of the HahnBanach theorem, proved in a non constructive way in [MP1]. In [CC] we give a direct constructive proof of HahnBanach theorem and using this, we can give a much simpler proof of the characterisation of compact convex sets. This note is organised as follows. First we recall the direct description the weak * topology on the unit ball of the dual of a normed space given in [CC]. We then use this to characterise the convex closed hull of a totally bounded subset. 1 Description of the unit ball in the dual space Let E be a normed vector space (over the rational numbers). We write x ∈ N(q) to express that the norm of x is < q. We describe a complete Heyting algebra by generators and relations. This complete Heyting algebra should be thought of as a pointfree description of of the unit ball of the dual E ′ for the weak * topology. The generators are formal expressions λx < q, with x ∈ E, and q ∈ Q and the relations are (1) [λx < q] ∧ [q < λx] = 0 (2) [λ(x + y) < r + s] ≤ [λx < r] ∨ [λy < s] (3) 1 = [λx < 1] if x ∈ N(1) where q < λx is defined to be λ(−x) < −q, together with the continuity axiom (4) [λx < q] = ∨q ′ <q[λx < q ′] Here are simple remarks about F n(E): Proposition 1.1 It follows from (1) and (2) that we have [λx < p] ∧ [λy < q] ≤ [λ(x + y) < p + q] If 0 < r we have [λ(rx) < rp] = [λx < p]. If r < s we have 1 = [λx < s] ∨ [r < λx]. If x ∈ N(q) we have 1 = [λx < q]. For any ɛ> 0 we have 1 = ∨q [q < λx] ∧ [λx < q + ɛ] Notice that we recover the generators λx ∈ (p, q) used in [MP1] by defining λx ∈ (p, q) =def [p < λx] ∧ [λx < q] 1 but the use of generators λx < q is the key to get a simple descrition of the frame F n(E) [CC]. In particular, [CC] provides a direct proof that this frame is compact regular, which is the localic form of Alaoglu’s theorem, and of the following result 1. Theorem 1.2 In F n(E) we have 1 = [λy0 < q] ∨... ∨ [λym−1 < q] if, and only if, there exists non negative rationals sj such that Σsj = 1 and Σsjyj ∈ N(q). 2 Convex
Minimal Invariant Spaces in Formal Topology
 The Journal of Symbolic Logic
, 1996
"... this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a ..."
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Cited by 4 (1 self)
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this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a pointfree formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a suitable formal topology, and the "existence" of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topological proof [6, 8], but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements. 1 Construction of Minimal Invariant Subspace
A constructive topological proof of van der Waerden's theorem
 Journal of Pure and Applied Algebra
, 1993
"... this paper was written, we became aware of the work [2, 15], which, in the quite different field of functional analysis, illustrates this common idea that localic methods can be used to find sharper reformulations of basic nonconstructive results, which become then constructively valid. Our work su ..."
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Cited by 3 (0 self)
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this paper was written, we became aware of the work [2, 15], which, in the quite different field of functional analysis, illustrates this common idea that localic methods can be used to find sharper reformulations of basic nonconstructive results, which become then constructively valid. Our work suggests that these methods, beside the purely mathematical advantage of solving problems concerned with equivariance or continuity in parameters [15], may be interesting also prooftheoretically in providing an elegant framework for extracting computational informations from given mathematical arguments. Our treatment of the topological proof of van der Waerden's theorem is apparently different from the one presented in Girard's book on proof theory [10]. We have not tried though to compare in detail the two arguments, because the main emphasis is somewhat different. The main points of our paper are the formulation of a pointfree version of a minimal property, whose ordinary version is proved via Zorn's lemma, and the observation that this pointfree version has a direct inductive proof