92 research outputs found
Coalgebras and Modal Expansions of Logics
AbstractIn this paper we construct a setting in which the question of when a logic supports a classical modal expansion can be made precise. Given a fully selfextensional logic S, we find sufficient conditions under which the Vietoris endofunctor V on S-referential algebras can be defined and we propose to define the modal expansions of S as the logic that arises from the V-coalgebras. As an example, we also show how the Vietoris endofunctor on referential algebras extends the Vietoris endofunctor on Stone spaces.From another point of view, we examine when a category of 'spaces' (X,A), ie sets X equipped with an algebra A of subsets of X, allows for the definition of powerspaces V (and hence transition systems (X,A)→V(X,A))
Heitman dimension of distributive lattices and commutative rings
This paper is the English translation of the first 4 sections of the article
"Thierry Coquand, Henri Lombardi, and Claude Quitt\'e. Dimension de Heitmann
des treillis distributifs et des anneaux commutatifs. Publications
Math\'ematiques de l'Universit\'e de Franche-Comt\'e, Besan\c{c}on. Alg\`ebre
et th\'eorie des nombres. (2006)", after some corrections.
Sections 5-7 of the original article are treated a bit more simply in the
book "Henri Lombardi and Claude Quitt\'e. Commutative algebra: constructive
methods. Finite projective modules. Algebra and applications, 20, Springer,
Dordrecht, 2015."
We study the notion of dimension introduced by Heitmann in his remarkable
article "Raymond Heitmann. Generating non-Noetherian modules efficiently, Mich.
Math. J., 31, (1084)" as well as a related notion, only implicit in his proofs.
We first develop this within the general framework of the theory of
distributive lattices and spectral spaces.
Keywords: Constructive mathematics, distributive lattice, Heyting algebra,
spectral space, Zariski lattice, Zariski spectrum, Krull dimension, maximal
spectrum, Heitmann lattice, Heitmann spectrum, Heitmann dimensions.
MSC: 13C15, 03F65, 13A15, 13E05Comment: A French translation follows the English version. arXiv admin note:
substantial text overlap with arXiv:1712.0195
04351 Abstracts Collection -- Spatial Representation: Discrete vs. Continuous Computational Models
From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351
``Spatial Representation: Discrete vs. Continuous Computational Models\u27\u27
was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Logical Berkovich Geometry: A Point-free Perspective
Extending our insights from \cite{NVOstrowski}, we apply point-free
techniques to sharpen a foundational result in Berkovich geometry. In our
language, given the ring of convergent power series
over a suitable non-Archimedean field , the points of its Berkovich Spectrum
correspond to -good filters. The surprise is
that, unlike the original result by Berkovich, we do not require the field
to be non-trivially valued. Our investigations into non-Archimedean geometry
can be understood as being framed by the question: what is the relationship
between topology and logic
Slanted canonicity of analytic inductive inequalities
We prove an algebraic canonicity theorem for normal LE-logics of arbitrary
signature, in a generalized setting in which the non-lattice connectives are
interpreted as operations mapping tuples of elements of the given lattice to
closed or open elements of its canonical extension. Interestingly, the
syntactic shape of LE-inequalities which guarantees their canonicity in this
generalized setting turns out to coincide with the syntactic shape of analytic
inductive inequalities, which guarantees LE-inequalities to be equivalently
captured by analytic structural rules of a proper display calculus. We show
that this canonicity result connects and strengthens a number of recent
canonicity results in two different areas: subordination algebras, and transfer
results via G\"odel-McKinsey-Tarski translations.Comment: arXiv admin note: text overlap with arXiv:1603.08515,
arXiv:1603.0834
Fibred contextual quantum physics
Inspired by the recast of the quantum mechanics in a toposical framework, we develop a contextual quantum mechanics via the geometric mathematics to propose a quantum contextuality adaptable in every topos.
The contextuality adopted corresponds to the belief that the quantum world must only be seen from the classical viewpoints à la Bohr consequently putting forth the notion of a context, while retaining a realist understanding. Mathematically, the cardinal object is a spectral Stone bundle Σ → B (between stably-compact locales) permitting a treatment of the kinematics, fibre by fibre and fully point-free. In leading naturally to a new notion of points, the geometricity permits to understand those of the base space B as the contexts C — the commutative C*–algebras of a incommutative C*–algebras — and those of the spectral locale Σ as the couples (C, ψ), with ψ a state of the system from the perspective of such a C. The contexts are furnished with a natural order, the aggregation order which is installed as the specialization on B and Σ thanks to (one part of) the Priestley's duality adapted geometrically as well as to the effectuality of the lax descent of the Stone bundles along the perfect maps
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