201 research outputs found
Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction
A (v,0)-semilattice is ultraboolean, if it is a directed union of finite
Boolean (v,0)-semilattices. We prove that every distributive (v,0)-semilattice
is a retract of some ultraboolean (v,0)-semilattices. This is established by
proving that every finite distributive (v,0)-semilattice is a retract of some
finite Boolean (v,0)-semilattice, and this in a functorial way. This result is,
in turn, obtained as a particular case of a category-theoretical result that
gives sufficient conditions, for a functor , to admit a right inverse. The
particular functor used for the abovementioned result about ultraboolean
semilattices has neither a right nor a left adjoint
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
On the commutativity of the powerspace constructions
We investigate powerspace constructions on topological spaces, with a
particular focus on the category of quasi-Polish spaces. We show that the upper
and lower powerspaces commute on all quasi-Polish spaces, and show more
generally that this commutativity is equivalent to the topological property of
consonance. We then investigate powerspace constructions on the open set
lattices of quasi-Polish spaces, and provide a complete characterization of how
the upper and lower powerspaces distribute over the open set lattice
construction
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