5,427 research outputs found
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
Uniqueness of directed complete posets based on Scott closed set lattices
In analogy to a result due to Drake and Thron about topological spaces, this
paper studies the dcpos (directed complete posets) which are fully determined,
among all dcpos, by their lattices of all Scott-closed subsets (such dcpos will
be called -unique).
We introduce the notions of down-linear element and quasicontinuous element
in dcpos, and use them to prove that dcpos of certain classes, including all
quasicontinuous dcpos as well as Johnstone's and Kou's examples, are
-unique. As a consequence, -unique dcpos with their
Scott topologies need not be bounded sober.Comment: 12 pages. arXiv admin note: substantial text overlap with
arXiv:1607.0357
Some applications of the ultrapower theorem to the theory of compacta
The ultrapower theorem of Keisler-Shelah allows such model-theoretic notions
as elementary equivalence, elementary embedding and existential embedding to be
couched in the language of categories (limits, morphism diagrams). This in turn
allows analogs of these (and related) notions to be transported into unusual
settings, chiefly those of Banach spaces and of compacta. Our interest here is
the enrichment of the theory of compacta, especially the theory of continua,
brought about by the immigration of model-theoretic ideas and techniques
Noncommutative Lattices and Their Continuum Limits
We consider finite approximations of a topological space by
noncommutative lattices of points. These lattices are structure spaces of
noncommutative -algebras which in turn approximate the algebra \cc(M) of
continuous functions on . We show how to recover the space and the
algebra \cc(M) from a projective system of noncommutative lattices and an
inductive system of noncommutative -algebras, respectively.Comment: 22 pages, 8 Figures included in the LaTeX Source New version, minor
modifications (typos corrected) and a correction in the list of author
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