Johnstone, P.T., Stone Spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we want to model. Section 4 describes our sheaf theoretic model; then geometric logic is used to test whether local properties can be lifted to a global level. 2 Preliminaries We present results from sheaf theory and geometric logic which we will use in our work. For de nitions we refer to [Joh82] or [MLM92] Notions from category theory and many sorted logic are assumed known. Categories and sheaves will usually be denoted in sans serif style, e.g. Set, Sh(I) Sheaf theory. Let I be a topological space, and (I) the topology on I . A presheaf on I is a functor P Set. Let U V be ....

....if for every S 1 ; S 2 2 InSys, their largest common t.c. subsystem S 1 S 2 is empty; the space is compact if additionally InSys is nite. In this situation a larger class of axioms is preserved by the global section functor (uniqueness in existential quanti cation is not required, cf. e.g. [Joh82], Ch. V.1.12) Then, the de nition of time as a sheaf N expresses the fact that independent systems may have independent clocks. 5 Conclusion We showed that a family InSys of interacting systems closed under pullbacks can be endowed with a topology which models the way these systems interact. ....

P. Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press, 1982.

....of functions over this space. This description is intuitively in term of observable quantities. Indeed, one primary source of motivation of these notes is in operator theory, where elements of an algebra represent quantities that can be observed. The approach of formal or point free topology [Menger, Johnstone] has also for aims to describe a space not in term of ideal points, but in term of observable notions. Thus we have two di erent ways of describing a space without using points, that are known classically to be equivalent. A natural question arises if the formal approach can be connected to ....

....and relations. In term of points, the points of Spec r (R) are the prime cone of R extending the given preorder [BCR] If the ring satis es some natural conditions considered in Stone s paper, we completely characterise the ordering of this lattice and we show that it is a normal lattice [Johnstone, CarralCoste], which means in term of points that any point is contained in a unique maximal point . We can then consider the maximal spectrum Max(R) associated to it. There is a natural map from R to C(Max(R) and we show constructively that, in a suitable sense, this map preserves the norm. This is one of ....

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P. T. Johnstone. Stone Spaces. Cambridge studies in advanced mathematics 3, 1982.

....formulae are automatically geometric. 7 Topological model In order to prove Theorem 6.2, we build a special topological model of the theory T . The domain M of this model is the set fa 0 ; a 1 ; g of all parameters. The truth values will form a complete Heyting algebra (or frame) H [6]. The model is de ned by choosing an interpretation [ F ] 2 H of each fact. Each closed conjunction C = F 1 Fm is then interpreted as usual by a nite meet [ C] F 1 ] F m ] each closed existential formula E = 9 x)C by an in nite join [ E] a M [ C[ x : ....

....join [ D] E 1 ] E n ] The Heyting algebra H and the interpretation [ will be built in such a way that we have F 1 ; Fm D i [ F 1 ] F m ] D] in H. Given that higher order intuitionistic logic can be interpreted in a topological model, see [6], Theorem 6.2 will follow at once. Following [6] we can de ne H by generators and relations. We take as generators the facts themselves, so that each closed conjunction C can be seen as an element C 2 H, and for the relations we take [ C 0 ] D 0 ] for each closed instance C 0 D 0 of ....

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P. Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982.

....would provide as well a constructive proof of the localic Steenrod s theorem and it may actually be interesting to formulate this theory in a predicative framework. 1. Formal Spaces and Distributive Lattices It will be convenient here to work with the following modi cation of the notion of site [8]. We represent a locale (or formal space) X as a distributive lattice D together with a nucleus j : Idl(D) Idl(D) on the frame Idl(D) of ideals of the lattice D. To give such a nucleus is equivalent to give a covering relation on D that is a relation a U between elements of D and ideals of D ....

Johnstone, P.T. Stone spaces. Cambridge Studies in Advanced Mathematics, 3. Cambridge University Press, Cambridge, 1986.

....enriched Dedekind MacNeille completion We will discuss an enriched version of the Dedekind MacNeille completion. This construction is presumably well known, but the author has not seen it written out in detail elsewhere. We remind ourselves about the construction in the pre order case (see e.g. Johnstone 82] p.109) Definition 3.17 Given a pre order A the MacNeille completion of A is the set M(A) of pairs (L,U) of subsets of A, such that L = lb(U) and U = ub(L) where lb(U) is the set of lower bounds of U and ub(L) is the set of upper bounds of L . The pre order on M(A) is (L1, U1) L2, U2) if ....

....same as a bimodule from 11 into A , and that a down subset is just a bimodule from A into 11 . Naturally we have [2 morphisms lb( A [2] A p [2] and ub( A p [2] A [2] for every A. Now we can define cuts k la Dedekind and MacNeille, essentially generalizing the presentation from [Johnstone 82] from pre orders (partial orders, actually) to enriched categories. Definition 3.20 A cut in a 2 category A is an up down subset (L, U) of A such that L = lb(U) and U = ub(L) Definition 3.21 The MacNeille completion of a [2 category A is the [2 category M(A) where M(A)o is the set of cuts in ....

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Johnstone, P.T., Stone Spaces, Cambridge Studies in Advanced Mathe- matics, vol. 3, Cambridge University Press, 1982.

....strongly pure followed by a weakly entire geometric morphism (whose domain is again an de nable dominance) Furthermore, if is a spread, then the factoriza 4 BUNGE FUNK JIBLADZE STREICHER tion is unique. This is a relative (or berwise) version of the pure entire factorization theorem given in [14]. For any (geometric) spread : F E over S [4] whose domain f : F S is a de nable dominance,it is shown in x4 that there is an inclusion F E H op over E , with H = f S . In x5 we carve out, by forcing methods [26] the largest subtopos E [H] of E H op containing F (over E) as a ....

....morphism of the diagonal : S , S S . Definition 2.2. An S Heyting algebra H in E is said to be an S Boolean algebra in E if the canonical morphism H : Part S (H) H de ned by H ( is an isomorphism. Recall that a locale A in a topos E is said to be a Stone locale [14] if it is a compact and zero dimensional locale in E . It is shown in [15] that A is a Stone locale if and only if it is is equivalent to one of the form Idl(B) for B a Boolean algebra. This equivalence holds because in this (the classical) case, B can be recovered as the sublattice (Idl(B) c ....

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P. T. Johnstone,Stone spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press(1982).

....directly by an analysis of the proof in [1] after a talk presenting some results of [5] As an application, we give a direct proof of Tychonov s theorem, which seems even more direct than the proof in [4] 1. Compact Space A space X will be given as a meet semi lattice with a covering relation [3] Cov(x) x 2 X. Each A 2 Cov(x) is a downward closed of X and we have yA 2 Cov(y) if y x. If U is a downward closed subset of X we define x U (read: U covers x) inductively by ffl x 2 U or ffl for some A 2 Cov(x) we have z U for all z 2 A: We say that U is a covering iff 1 U iff x U ....

Johnstone, Peter T. Stone spaces. Cambridge Studies in Advanced Mathematics, 3. Cambridge University Press, Cambridge, 1986.

....on these types; PCF can be considered as the theoretical model for functional programming languages. Domain theory has developed extensively in the past three decades and is now a major paradigm in the semantics of programming languages. For a basic introduction to its theory and applications, see [59, 74, 96, 64, 2, 114]. Algebraic domains have also been used to represent classical spaces in mathematics in an e#ective framework. Weihrauch and Schreiber [124] constructed an embedding of a Polish space (a topologically complete separable metrizable space) into an algebraic domain. Stoltenberg Hansen and Tucker have ....

P. T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982.

....Preliminary version at September 22, 2000 Introduction Stone duality between Boolean algebra and compact totally disconnected spaces provides an algebraic and point free 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 al..l 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 ....

....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 preorder so ....

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P. T. Johnstone. Stone Spaces. Cambridge studies in advanced mathematics 3, 1982.

.... are handled more easily; it allows for more natural categorical explanations (e.g. by means of adjunctions) and it is more applicable to computer science situations (where relative information preservation is important) For references to the covariant form see [7] 51] 9] 32] 25] [35], 15] 42] 31] and [19] 1 Galois main results were published fourteen years after his early death (at the age of 21 in a duel) by Liouville in his Journal de math ematiques pures et appliqu ees (1846) For a translation of Galois original notes Memoire sur les conditions de ....

....with the duality above, one arrives at a nice symmetric duality between the categories Z SZ 0 and Z 0 SZ . Now, special choices of Z and Z 0 and suitable restrictions provide a multitude of classical dualities, encompassing the dualities between ffl sober spaces and spatial frames [25] [35] ffl sober spaces with compact open base and distributive semilattices [28] ffl Boolean spaces ( Stone spaces) and Boolean lattices [54] 35] ffl semilattices and algebraic lattices [24] 32] ffl Alexandroff spaces ( posets) and completely distributive algebraic lattices [16] and many ....

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Johnstone, P. Stone Spaces. No. 3 in Cambridge Studies in advanced Mathematics. Cambridge University Press, Cambridge, UK, 1982.

.... are handled more easily; it allows for more natural categorical explanations (e.g. by means of adjunctions) and it is more applicable to computer science situations (where relative information preservation is important) For references to the covariant form see [6] 50] 8] 31] 24] [34], 14] 41] 30] and [18] 1 Galois main results were published fourteen years after his early death (at the age of 21 in a duel) by Liouville in his Journal de math ematiques pures et appliqu ees (1846) For a translation of Galois original notes Memoire sur les conditions de ....

....the duality above, one arrives at a nice symmetric duality between the categories Z SZ 0 and Z 0 SZ . Now, special choices of Z and Z 0 and suitable restrictions provide a multitude of classical dualities, encompassing the dualities between ffl sober spaces and spatial frames [24] [34] ffl sober spaces with compact open base and distributive semilattices [27] ffl Boolean spaces ( Stone spaces) and Boolean lattices [53] 34] ffl semilattices and algebraic lattices [23] 31] ffl Alexandroff spaces ( posets) and completely distributive algebraic lattices [15] and many ....

[Article contains additional citation context not shown here]

Johnstone, P. Stone Spaces. No. 3 in Cambridge Studies in advanced Mathematics. Cambridge University Press, Cambridge, UK, 1982.

....enriched Dedekind MacNeille completion We will discuss an enriched version of the Dedekind MacNeille completion. This construction is presumably well known, but the author has not seen it written out in detail elsewhere. We remind ourselves about the construction in the pre order case (see e.g. Johnstone 82] p.109) Definition 3.17 Given a pre order A the MacNeille completion of A is the set M(A) of pairs (L; U) of subsets of A, such that L = lb(U) and U = ub(L) where lb(U) is the set of lower bounds of U and ub(L) is the set of upper bounds of L . The pre order on M(A) is (L 1 ; U 1 ) L 2 ; U ....

.... Omega morphisms lb( A Gammaffl Omega Omega ] Gammaffl [A op Gammaffl Omega Omega ] and ub( A op Gammaffl Omega Omega ] Gammaffl [A Gammaffl Omega Omega ] for every A. Now we can define cuts a la Dedekind and MacNeille, essentially generalizing the presentation from [Johnstone 82] from pre orders (partial orders, actually) to enriched categories. Definition 3.20 A cut in a Omega Omega category A is an up down subset (L; U) of A such that L = lb(U) and U = ub(L) 2 Definition 3.21 The MacNeille completion of a Omega Omega category A is the Omega Omega category ....

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Johnstone, P.T., Stone Spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, 1982.

....spatial of the generalized spaces. They are the toposes that are determined by the subobjects of 1 with no need to consider the rest of the G frame. The tendency in topos theory is to study locales in place of topological spaces, and you can read more about them in Vickers [6] and Johnstone [3]. 5 Summary A topos as generalized space is the space of models for a geometric theory. The whole story is in answer to the question of what space means here. If the theory is presented as T , then its topos its classifying topos is denoted [T ] The points of [T ] are the models of ....

Peter T. Johnstone. Stone Spaces. Vol. 3, Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1982.

....is defined at the level of powerset l.s.p. s, as before, and then also generalized to the algebraic setting. Although worlds seem to play an essential role in the construction, and recovering worlds out of abstract algebras would require the use of Stone like representation techniques [31] that are, at least for the moment, out of our scope of investigation, we can still manage to find the appropriate extension of the construction. 3.1 Synchronization As explained before, synchronization is a very simple mechanism of combination that has been studied in [42, 43] It is the ....

P. Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1982.

....have 1 Delta L in M I : In particular, we get [8y 2 L:Z I (y) Z I (1) and hence Z I (1) This implies OE [1 = 0] by definition of I ; and hence OE if B is a consistent boolean algebra. 1. 8 Generalisation to Stone spaces All these constructions can be generalised to the case of Stone spaces [10] 1 , that can be described as spaces of prime filters of a distributive lattice. Given such a lattice D; and a lattice morphism g; we can associate the formal propositional theory 1. 1) 2. 0) 3. if (x) and x y; then (y) 4. if (x) and (y) then (x:y) 5. if (x) then ....

....without points is used to give a theory entirely parallel to Gelfand s, such that it is possible at every stage to reach the corresponding stage in Gelfand s theory by a simple application of the axiom of choice [13] All we shall use of the previous sections is proposition 1. 1 Johnstone [10] call these spaces coherent spaces. 2.1.1 General Notations We recall first some terminology extracted from [3] A block is a finite sequence of 0s and 1s. We use the notation A; B; C; for blocks, and write AB for the concatenation of two blocks A and B. If A = b 1 : b p then p is ....

P. Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics, 1981

....the network. 5.1.4 Conclusion The upshot of these observations is that exponentiation can be seen as an operation taking a schedule and yielding an abstract machine or automaton for executing the events of the schedule. This state of affairs is strongly reminiscent of Stone duality (see Johnstone [26]) It would be very interesting to investigate this connection further. 5.2 Choice Although the structures studied in this thesis were suggested by application to the semantics of concurrency, I have presented only a part of the required theory. A full model of concurrency must incorporate a ....

Peter T. Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics 3. Cambridge University Press, 1982.

....that does not use this axiom. It does not seem, however, that this method has been used in extracting constructive informations from given concrete examples of a complex proof of an arithmetical statement. Stimulated by results of de Bruijn, van der Meiden [3] of Martin Lof [14] and of Johnstone [13], we started in [5] a program of constructivisation of some infinitary proofs in combinatorics (Ramsey s theorem, Higman s lemma) by applying localic methodology. Although the extent to which this yields new results is not yet known, the ideas of point free topology seem to provide intuitions into ....

P. Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics, 1981

....is proved in [4] but with a proof that uses classical reasoning 1 . Our arguments are valid both in topos theory and in a predicative theory such as CZF [1] 1. Formal Spaces and Distributive Lattices It will be convenient here to work with the following modification of the notion of site [5]. We represent a formal space X as a distributive lattice D together with a nucleus j : Idl(D) Idl(D) on the complete Heyting algebra Idl(D) of ideals of the lattice D, where an ideal of D is a subset U D such that ffl 0 2 U ffl x 2 U if y 2 U and x y ffl x 1 x 2 2 U if x 1 ; x 2 2 U ....

Johnstone, Peter T. Stone spaces. Cambridge Studies in Advanced Mathematics, 3. Cambridge University Press, Cambridge, 1986.

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Johnstone, P.T., Stone Spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, 1982.

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P. T. Johnstone. Stone Spaces. Cambridge studies in advanced mathematics 3, 1982.

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P. Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982.

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Johnstone, P.J., Stone Spaces, Cambridge Studies in Advanced Mathematics, 1981.

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Peter T. Johnstone. Stone Spaces, Cambridge Studies in Advanced Mathematics, Vol. 3, 1981.

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Johnstone, P. T. Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982.

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P. T. Johnstone, Stone Spaces. Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, 1982.

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