838 research outputs found
Non-principal ultrafilters, program extraction and higher order reverse mathematics
We investigate the strength of the existence of a non-principal ultrafilter
over fragments of higher order arithmetic.
Let U be the statement that a non-principal ultrafilter exists and let
ACA_0^{\omega} be the higher order extension of ACA_0. We show that
ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that
ACA_0^{\omega}+\U is conservative over PA.
Moreover, we provide a program extraction method and show that from a proof
of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U
a realizing term in G\"odel's system T can be extracted. This means that one
can extract a term t, such that A(f,t(f))
A non-commutative generalization of Stone duality
We prove that the category of boolean inverse monoids is dually equivalent to
the category of boolean groupoids. This generalizes the classical Stone duality
between boolean algebras and boolean spaces. As an instance of this duality, we
show that the boolean inverse monoid associated with the Cuntz groupoid is the
strong orthogonal completion of the polycyclic (or Cuntz) monoid and so its
group of units is a Thompson group
Ultrafilter spaces on the semilattice of partitions
The Stone-Cech compactification of the natural numbers bN, or equivalently,
the space of ultrafilters on the subsets of omega, is a well-studied space with
interesting properties. If one replaces the subsets of omega by partitions of
omega, one can define corresponding, non-homeomorphic spaces of partition
ultrafilters. It will be shown that these spaces still have some of the nice
properties of bN, even though none is homeomorphic to bN. Further, in a
particular space, the minimal height of a tree pi-base and P-points are
investigated
Reduced Coproducts of Compact Hausdorff Spaces
By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the reduced coproduct , which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the ultracoproduct can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces
Survey on the Tukey theory of ultrafilters
This article surveys results regarding the Tukey theory of ultrafilters on
countable base sets. The driving forces for this investigation are Isbell's
Problem and the question of how closely related the Rudin-Keisler and Tukey
reducibilities are. We review work on the possible structures of cofinal types
and conditions which guarantee that an ultrafilter is below the Tukey maximum.
The known canonical forms for cofinal maps on ultrafilters are reviewed, as
well as their applications to finding which structures embed into the Tukey
types of ultrafilters. With the addition of some Ramsey theory, fine analyses
of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page
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