114,093 research outputs found
Ultrafilter extensions of linear orders
It was recently shown that arbitrary first-order models canonically extend to
models (of the same language) consisting of ultrafilters. The main precursor of
this construction was the extension of semigroups to semigroups of
ultrafilters, a technique allowing to obtain significant results in algebra and
dynamics. Here we consider another particular case where the models are
linearly ordered sets. We explicitly calculate the extensions of a given linear
order and the corresponding operations of minimum and maximum on a set. We show
that the extended relation is not more an order however is close to the natural
linear ordering of nonempty half-cuts of the set and that the two extended
operations define a skew lattice structure on the set of ultrafilters
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Formal concept analysis has grown from a new branch of the mathematical field
of lattice theory to a widely recognized tool in Computer Science and
elsewhere. In order to fully benefit from this theory, we believe that it can
be enriched with notions such as approximation by computation or
representability. The latter are commonly studied in denotational semantics and
domain theory and captured most prominently by the notion of algebraicity, e.g.
of lattices. In this paper, we explore the notion of algebraicity in formal
concept analysis from a category-theoretical perspective. To this end, we build
on the the notion of approximable concept with a suitable category and show
that the latter is equivalent to the category of algebraic lattices. At the
same time, the paper provides a relatively comprehensive account of the
representation theory of algebraic lattices in the framework of Stone duality,
relating well-known structures such as Scott information systems with further
formalisms from logic, topology, domains and lattice theory.Comment: 36 page
Causal sites as quantum geometry
We propose a structure called a causal site to use as a setting for quantum
geometry, replacing the underlying point set. The structure has an interesting
categorical form, and a natural "tangent 2-bundle," analogous to the tangent
bundle of a smooth manifold. Examples with reasonable finiteness conditions
have an intrinsic geometry, which can approximate classical solutions to
general relativity. We propose an approach to quantization of causal sites as
well.Comment: 21 pages, 3 eps figures; v2: added references; to appear in JM
A domain of spacetime intervals in general relativity
Beginning from only a countable dense set of events and the causality
relation, it is possible to reconstruct a globally hyperbolic spacetime in a
purely order theoretic manner. The ultimate reason for this is that globally
hyperbolic spacetimes belong to a category that is equivalent to a special
category of domains called interval domains.Comment: 25 page
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