8,058 research outputs found
Constructive version of Boolean algebra
The notion of overlap algebra introduced by G. Sambin provides a constructive
version of complete Boolean algebra. Here we first show some properties
concerning overlap algebras: we prove that the notion of overlap morphism
corresponds classically to that of map preserving arbitrary joins; we provide a
description of atomic set-based overlap algebras in the language of formal
topology, thus giving a predicative characterization of discrete locales; we
show that the power-collection of a set is the free overlap algebra
join-generated from the set. Then, we generalize the concept of overlap algebra
and overlap morphism in various ways to provide constructive versions of the
category of Boolean algebras with maps preserving arbitrary existing joins.Comment: 22 page
On some peculiar aspects of the constructive theory of point-free spaces
This paper presents several independence results concerning the topos-valid
and the intuitionistic (generalized) predicative theories of locales. In
particular, certain consequences of the consistency of a general form of
Troelstra's uniformity principle with constructive set theory and type theory
are examined
The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach
We compare two influential ways of defining a generalized notion of space.
The first, inspired by Gelfand duality, states that the category of
'noncommutative spaces' is the opposite of the category of C*-algebras. The
second, loosely generalizing Stone duality, maintains that the category of
'pointfree spaces' is the opposite of the category of frames (i.e., complete
lattices in which the meet distributes over arbitrary joins). One possible
relationship between these two notions of space was unearthed by Banaschewski
and Mulvey, who proved a constructive version of Gelfand duality in which the
Gelfand spectrum of a commutative C*-algebra comes out as a pointfree space.
Being constructive, this result applies in arbitrary toposes (with natural
numbers objects, so that internal C*-algebras can be defined). Earlier work by
the first three authors, shows how a noncommutative C*-algebra gives rise to a
commutative one internal to a certain sheaf topos. The latter, then, has a
constructive Gelfand spectrum, also internal to the topos in question. After a
brief review of this work, we compute the so-called external description of
this internal spectrum, which in principle is a fibered pointfree space in the
familiar topos Sets of sets and functions. However, we obtain the external
spectrum as a fibered topological space in the usual sense. This leads to an
explicit Gelfand transform, as well as to a topological reinterpretation of the
Kochen-Specker Theorem of quantum mechanics, which supplements the remarkable
topos-theoretic version of this theorem due to Butterfield and Isham.Comment: 12 page
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Overlap Algebras as Almost Discrete Locales
Boolean locales are almost discrete. In fact, spatial Boolean locales are the
same thing as discrete spaces. This does not make sense intuitionistically,
since (non-trivial) discrete locales fail to be Boolean. We show that Sambin's
"overlap algebras" have good enough features to be called "almost discrete
locales".
Keywords. Strongly dense sublocales, almost discrete spaces, overlap
algebras, constructive topology
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
Overlap Algebras: a Constructive Look at Complete Boolean Algebras
The notion of a complete Boolean algebra, although completely legitimate in
constructive mathematics, fails to capture some natural structures such as the
lattice of subsets of a given set. Sambin's notion of an overlap algebra,
although classically equivalent to that of a complete Boolean algebra, has
powersets and other natural structures as instances. In this paper we study the
category of overlap algebras as an extension of the category of sets and
relations, and we establish some basic facts about mono-epi-isomorphisms and
(co)limits; here a morphism is a symmetrizable function (with classical logic
this is just a function which preserves joins). Then we specialize to the case
of morphisms which preserve also finite meets: classically, this is the usual
category of complete Boolean algebras. Finally, we connect overlap algebras
with locales, and their morphisms with open maps between locales, thus
obtaining constructive versions of some results about Boolean locales.Comment: Postproceedings of CCC2018: Continuity, Computability,
Constructivity. Faro, Portugal, 24-28 Sep 201
Information completeness in Nelson algebras of rough sets induced by quasiorders
In this paper, we give an algebraic completeness theorem for constructive
logic with strong negation in terms of finite rough set-based Nelson algebras
determined by quasiorders. We show how for a quasiorder , its rough
set-based Nelson algebra can be obtained by applying the well-known
construction by Sendlewski. We prove that if the set of all -closed
elements, which may be viewed as the set of completely defined objects, is
cofinal, then the rough set-based Nelson algebra determined by a quasiorder
forms an effective lattice, that is, an algebraic model of the logic ,
which is characterised by a modal operator grasping the notion of "to be
classically valid". We present a necessary and sufficient condition under which
a Nelson algebra is isomorphic to a rough set-based effective lattice
determined by a quasiorder.Comment: 15 page
- …