316 research outputs found
Completion of continuity spaces with uniformly vanishing asymmetry
The classical Cauchy completion of a metric space (by means of Cauchy
sequences) as well as the completion of a uniform space (by means of Cauchy
filters) are well-known to rely on the symmetry of the metric space or uniform
space in question. For qausi-metric spaces and quasi-uniform spaces various
non-equivalent completions exist, often defined on a certain subcategory of
spaces that satisfy a key property required for the particular completion to
exist. The classical filter completion of a uniform space can be adapted to
yield a filter completion of a metric space. We show that this completion by
filters generalizes to continuity spaces that satisfy a form of symmetry which
we call uniformly vanishing asymmetry
Bohrification of operator algebras and quantum logic
Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems. We show that a possible
resolution of these difficulties, suggested by the ideas of Bohr, emerges if
instead of single projections one considers elementary propositions to be
families of projections indexed by a partially ordered set C(A) of appropriate
commutative subalgebras of A. In fact, to achieve both maximal generality and
ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart
C*-subalgebras of A. Such families of projections form a Heyting algebra in a
natural way, so that the associated propositional logic is intuitionistic:
distributivity is recovered at the expense of the law of the excluded middle.
Subsequently, generalizing an earlier computation for n-by-n matrices, we
prove that the Heyting algebra thus associated to A arises as a basis for the
internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the
"Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of
functors from C(A) to the category of sets. We explain the relationship of this
construction to partial Boolean algebras and Bruns-Lakser completions. Finally,
we establish a connection between probability measure on the lattice of
projections on a Hilbert space H and probability valuations on the internal
Gelfand spectrum of A for A = B(H).Comment: 31 page
Going down in (semi)lattices of finite Moore families and convex geometries
International audienceIn this paper we first study the changes occuring in the posets of irreducible elements when one goes from an arbitrary Moore family (respectively, a convex geometry) to one of its lower covers in the lattice of all Moore families (respectively, in the semilattice of all convex geometries) defined on a finite set. Then, we show that the poset of all convex geometries that have the same poset of join-irreducible elements is a ranked join-semilattice, and we give an algorithm for computing it. Finally, we prove that the lattice of all ideals of a given poset P is the only convex geometry having a poset of join-irreducible elements isomorphic to P if and only if the width of P is less than 3.Dans ce texte, nous Ă©tudions d'abord les changements dans les ensembles ordonnĂ©s d'Ă©lĂ©ments irrĂ©ductibles lorsqu'on passe d'une famille de Moore arbitraire (respectivement, d'une gĂ©omĂ©trie convexe) Ă l'une de ses couvertures infĂ©rieures dans le treillis de toutes les familles de Moore (respectivement, dans le demi-treillis des gĂ©omĂ©tries convexes). Nous montrons ensuite que l'ensemble ordonnĂ© de toutes les gĂ©omĂ©tries convexes ayant le mĂȘme ensemble ordonnĂ© d'Ă©lĂ©ments sup-irrĂ©ductibles est un demi-treillis rangĂ© et nous donnons un algorithme pour le calculer. Enfin nous caractĂ©risons les ensembles ordonnĂ©s P pour lesquels le treillis de leurs idĂ©aux est l'unique gĂ©omĂ©trie convexe ayant son ensemble ordonnĂ© d'Ă©lĂ©ments sup-irrĂ©ductibles isomorphe Ă P
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