## Introduction About Stone’s notion of Spectrum

### BibTeX

@MISC{Coquand_introductionabout,

author = {Thierry Coquand},

title = {Introduction About Stone’s notion of Spectrum},

year = {}

}

### OpenURL

### Abstract

The goal of this paper is to analyse two remarkable notes by Stone [StoI, StoII]. Both describe a compact space in term of some algebra of functions over this space. This description

### Citations

433 | Constructive Analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...As a typical example Segal’s notion of integration algebra [Seg] is expressed in our framework. We show then that in some cases, we can compute effectively the points of the maximal spectrum, like in =-=[Bis]-=-. Finally we explain what happens to the case of f-rings, a structure that combines the two structures considered by Stone. Most results in this paper are elementary results about distributive lattice... |

165 |
Géométrie Algébrique Réelle
- Bochnak, Coste, et al.
- 1987
(Show Context)
Citation Context ...l spectrum which is here defined as a distributive lattice Specr(R) given by generators and relations. (In term of points, the points of Specr(R) are the prime cones of R extending the given preorder =-=[BCR]-=-.) If the ring satisfies 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 [Joh, CaC], which means in... |

71 |
Lattice Theory", Third Edition
- Birkhoff
- 1967
(Show Context)
Citation Context ...lassical logic and the axiom of choice however, we know that Max(R) being compact regular, has enough points [Joh]. 17s6 f-ring The structure of f-ring combines the two structures considered by Stone =-=[Bir]-=-. We consider only the case where we have a strong unit 1, in which case the structure can be simply described as an ordered ring which has also a binary sup operation. A typical example is provided b... |

34 |
Anneaux préordonnés
- Krivine
- 1964
(Show Context)
Citation Context ...a suitable sense, this map preserves the norm. This is one of the main point of Gelfand duality, which is proved non constructively in [Joh] and [BM1] 2 . We show in this way that the main results in =-=[Kri]-=- have natural constructive proofs 3 . In particular, we obtain constructive proofs of theorems such as Kadison-Dubois [BS] by reading Krivine’s arguments in a point-free setting. We then give a simila... |

33 | Inductively generated formal topologies
- Coquand, Sambin, et al.
- 2003
(Show Context)
Citation Context ...eorems. We present some theorems in the framework of the theory of locales [Joh] but it can be worth noting that they can be formulated as well in the predicative framework of formal topology instead =-=[Sam]-=-. 1 First representation theorem The goal of this section is to show a representation theorem, which gives a way to represent the elements f of an ordered ring R as continuous functions over a compact... |

27 | Dynamical method in algebra: Effective Nullstellensätze - Coste, Lombardi, et al. - 2001 |

20 |
Spaces, Cambridge studies in advanced mathematics 3
- Johnstone, Stone
- 1983
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Citation Context ...ural 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 the main point of Gelfand duality, which is proved non constructively in =-=[Joh]-=- and [BM1] 2 . We show in this way that the main results in [Kri] have natural constructive proofs 3 . In particular, we obtain constructive proofs of theorems such as Kadison-Dubois [BS] by reading K... |

20 | A globalization of the Hahn-Banach theorem - Mulvey, Pelletier - 1991 |

17 |
Completeness and axiomatizability
- Scott
- 1971
(Show Context)
Citation Context ...uality D(a1) ∧ . . . ∧ D(ak) ∧ D(−p) � D(b1) ∨ . . . ∨ D(bl) The next fundamental Theorem states that this way of proving inequality is complete. (This is essentially a form of cut-elimination result =-=[Sco]-=-.) Theorem 1.8 We have D(a1) ∧ . . . ∧ D(ak) � D(b1) ∨ . . . ∨ D(bl) in Specr(R) iff we have a relation m + p = 0 where m belongs to the multiplicative monoid generated by a1, . . . , ak and p belongs... |

13 |
Le théorème de Chevalley-Tarski. Cahiers de Topologie et Géométrie Différentielle 16
- Joyal
- 1975
(Show Context)
Citation Context ...+ s) and D(−a − s) = D(−a + s) ∧ D(−a − s) and hence D(a 2 − s 2 ) = D(a − s) ∨ D(−a − s) 4sThe definition of Specr(R) should be compared to Joyal’s point-free definition of the Zariski spectrum of R =-=[Joy]-=-, seen as a ring, which is defined as the distributive lattice generated by the symbols I(a), a in R and the axioms I(0) = 0 I(a + b) � I(a) ∨ I(b) I(1) = 1 I(ab) = I(a) ∧ I(b) These axioms are satisf... |

12 | Normal spectral spaces and their dimension - Carral, Coste - 1983 |

12 |
Entailment Relations and Distributive Lattices. Proceeding of Logic Colloquium
- Coquand
- 1998
(Show Context)
Citation Context ...I(a + b) � I(a) ∨ I(b) I(1) = 1 I(ab) = I(a) ∧ I(b) These axioms are satisfied if we interpret I(a) as D(a) ∨ D(−a) in Specr(R) 6 . We shall need the following characterisation of Specr(R), stated in =-=[CC]-=-, which holds more generally for all commutative rings R with a preorder such that all square are positive, but not necessarily divisible or archimedean. This is essentially a version of the formal Po... |

12 |
Algebraic integration theory
- Segal
(Show Context)
Citation Context ...s Kadison-Dubois [BS] by reading Krivine’s arguments in a point-free setting. We then give a similar treatment for the second note of Stone. As a typical example Segal’s notion of integration algebra =-=[Seg]-=- is expressed in our framework. We show then that in some cases, we can compute effectively the points of the maximal spectrum, like in [Bis]. Finally we explain what happens to the case of f-rings, a... |

10 |
Quelques propriétés des préordres dans les anneaux commutatifs unitaires
- Krivine
- 1964
(Show Context)
Citation Context ... of Kadison-Dubois is actually more general in that it does not assume that P contains all squares. In this subsection we show how to deal with this generalisation, following and simplifying slightly =-=[Kr1]-=-. Lemma 1.18 For all n we can write x 2 + 1 = P (n − x, n + x) where P (X, Y ) is a rational homogeneous polynomial with coefficients � 0. Proof. We use the change of variables y(n + x) = n − x. The q... |

10 |
A general theory of spectra
- STONE
- 1941
(Show Context)
Citation Context ...y be ambiguous; however in practice this ambiguity is not a problem since it is always clear from the context if we mean r in Q or the element r.1 in R.) The elements of R are thought of as operators =-=[StoI]-=- and the elements of P are the positive operators. The relation a � b defined as b − a ∈ P is a preorder on R such that 0 � a 2 for all a in R. The ring R is in particular a predordered vector space o... |

9 |
Zum Darstellungssatz von Kadison-Dubois
- BECKER, SCHWARTZ
(Show Context)
Citation Context ...uctively in [Joh] and [BM1] 2 . We show in this way that the main results in [Kri] have natural constructive proofs 3 . In particular, we obtain constructive proofs of theorems such as Kadison-Dubois =-=[BS]-=- by reading Krivine’s arguments in a point-free setting. We then give a similar treatment for the second note of Stone. As a typical example Segal’s notion of integration algebra [Seg] is expressed in... |

9 |
A constructive proof of the Stone-Weierstrass theorem
- Banaschewski, Mulvey
- 1997
(Show Context)
Citation Context ...lso f ∧g ∈ V ) and the collection of open sets D(f) = f −1 (0, ∞) form a basis for the topology of X. The next proposition states the existence of partition of unity, without having to mention points =-=[BM3]-=-. Proposition 3.1 If Uj is an arbitrary covering of X it is possible to find a partition of unity p1, . . . , pn with pi ∈ V, 0 � pi � 1 and Σpi = 1 and each open D(pi) is a formal subset of some Uj. ... |

7 |
Open locales and exponentiation. In Mathematical applications of category theory
- Johnstone
- 1983
(Show Context)
Citation Context ...ws in such a case how to build effectively some points of Max(R), using dependent choice. 5 Positivity on Max(R) We state first a general result on compact completely regular locales. We refer to the =-=[JoO]-=- for a definition of open locales. Intuitively, it means that we have a predicate on open subsets, called positivity predicate, which expresses when an open is inhabited 8 . Theorem 5.1 If X is a comp... |

6 | Topology without points - Menger - 1978 |

6 |
Algebraische fassung des massproblems
- Tarski
- 1938
(Show Context)
Citation Context ...ace Max(R) is the Stone dual space of B. Proof. In this case Max(R) coincides with the spectral frame defined by Specr(R) and Specr(R) coincides with B. The construction of this ring R is implicit in =-=[Tar]-=-, and is useful for analysing measures on B. This is because v:B → R is the universal valuation. If w:B → S is another valuation in an ordered Q-vector space S, with a distinguished positive element 1... |

5 |
The spectral theory of commutative C ∗ -algebras: the constructive Gelfand-Mazur theorem, Quaest
- Banaschewski, Mulvey
(Show Context)
Citation Context ...rom R to C(Max(R)) and we show constructively that, in a suitable sense, this map preserves the norm. This is one of the main point of Gelfand duality, which is proved non constructively in [Joh] and =-=[BM1]-=- 2 . We show in this way that the main results in [Kri] have natural constructive proofs 3 . In particular, we obtain constructive proofs of theorems such as Kadison-Dubois [BS] by reading Krivine’s a... |

5 |
Sur la décomposition des opérations fonctionelles linéaires
- Riesz
- 1930
(Show Context)
Citation Context ...ex” framework presentation of Gelfand duality. 3 The space Max(R) is called Sp(R) in Krivine’s paper [Kri], which does not consider the space corresponding to Specr(R). 1sfurther the insight of Riesz =-=[Rie]-=- and Stone that some basic results in functional analysis can be captured by simple algebraic statements. The versions “without points” of the various representation theorems that we present imply dir... |

3 | Algèbre, chapitre 6 - Bourbaki - 1964 |

2 | Positive polynomials on compact sets - Wormann - 2001 |