2,026 research outputs found
Quasi-selective ultrafilters and asymptotic numerosities
We isolate a new class of ultrafilters on N, called "quasi-selective" because
they are intermediate between selective ultrafilters and P-points. (Under the
Continuum Hypothesis these three classes are distinct.) The existence of
quasi-selective ultrafilters is equivalent to the existence of "asymptotic
numerosities" for all sets of tuples of natural numbers. Such numerosities are
hypernatural numbers that generalize finite cardinalities to countable point
sets. Most notably, they maintain the structure of ordered semiring, and, in a
precise sense, they allow for a natural extension of asymptotic density to all
sequences of tuples of natural numbers.Comment: 27 page
Ultrafilters maximal for finite embeddability
In [1] the authors showed some basic properties of a pre-order that arose in
combinatorial number theory, namely the finite embeddability between sets of
natural numbers, and they presented its generalization to ultrafilters, which
is related to the algebraical and topological structure of the Stone-\v{C}ech
compactification of the discrete space of natural numbers. In this present
paper we continue the study of these pre-orders. In particular, we prove that
there exist ultrafilters maximal for finite embeddability, and we show that the
set of such ultrafilters is the closure of the minimal bilateral ideal in the
semigroup (\bN,\oplus), namely \overline{K(\bN,\oplus)}. As a consequence,
we easily derive many combinatorial properties of ultrafilters in
\overline{K(\bN,\oplus)}. We also give an alternative proof of our main
result based on nonstandard models of arithmetic
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
Quasi-selective ultrafilters and asymptotic numerosities
We isolate a new class of ultrafilters on N, called “quasi-selective” because they are intermediate between selective ultrafilters and P-points. (Under the Continuum Hypothesis these three classes are distinct.) The existence of quasi-selective ultrafilters is equivalent to the existence of “asymptotic numerosities” for all sets of tuples A ⊆ N^k. Such numerosities are hypernatural numbers that generalize finite cardinalities to countable point sets. Most notably, they maintain the structure of ordered semiring, and, in a precise sense, they allow for a natural extension of asymptotic density to all sets of tuples of natural numbers
Finite Embeddability of Sets and Ultrafilters
A set A of natural numbers is finitely embeddable in another such set B if
every finite subset of A has a rightward translate that is a subset of B. This
notion of finite embeddability arose in combinatorial number theory, but in
this paper we study it in its own right. We also study a related notion of
finite embeddability of ultrafilters on the natural numbers. Among other
results, we obtain connections between finite embeddability and the algebraic
and topological structure of the Stone-Cech compactification of the discrete
space of natural numbers. We also obtain connections with nonstandard models of
arithmetic.Comment: to appear in Bulletin of the Polish Academy of Sciences, Math Serie
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