Questions tagged [ultrafilters]
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A nonprincipal ultrafilter that is not a $p$-point
On pg 76 of Jech's Set Theory, he proves the existence of a nonprincipal ultrafilter on $\omega$ that is not a $p$-point.
Given a partition $\{A_n\}$ of $\omega$ into $\aleph_0$ infinite pieces, ...
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Pseudo-intersections, splitting families, and ultrafilters
Suppose $U$ is a non-principal ultrafilter on $\omega$, and let us define $\tau(U)$ to be the minimum cardinality of a family $\mathcal{X}\subseteq U$ such that $\mathcal{X}$ does not have an infinite ...
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Is the product of commuting ultrafilters an ultrafilter?
If $U$ is a filter on $X$ and $W$ is a filter on $Y$, their product is the filter $U\times W$ on $X\times Y$ generated by rectangles $A\times B$ where $A\in U$ and $B\in W$.
In certain circumstances ...
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A question on simple $P_{\aleph_2}$-points
This question is motivated by discussion surrounding this MO question.
An ultrafilter $U$ on $\omega$ is a simple $P_{\aleph_2}$-point if it is generated by a sequence $\langle X_\alpha:\alpha<\...
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The existence of $T$-ultrafilters in ZFC
Looking at this MO-problem, my collegue Igor Protasov suggested to ask on Mathoverflow his old question on $T$-ultrafilters hoping that somebody on MO can solve it.
First I recall the necessary ...
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Ultrafilter on the ordinal $\omega^\omega$
For any ultrafilter $\mathcal{U}$ on $\omega$ and any finite $k$ we can construct tensor power $\mathcal{U}^{\otimes k}$ which is ultrafilter on $\omega^k$. Does there exist some natural extension of ...
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The intersection of all normal ultrafilters on a measurable cardinal
Suppose $\kappa$ is a measurable cardinal. Let $W$ be the intersection of all normal ultrafilters on $\kappa$.
I am interested in a precise characterization of the filter $W$.
One sure way to ...
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Club filter basis in $\omega_1$
My question is about existing of basis of club filter club($\omega_1$) with cardinality $c$. Does it exist?
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Ultrapower of amenable group
Let $\Gamma$ be an amenable group. Consider its ultrapower $^*\Gamma$. It is known that $^*\Gamma$ need not be amenable. In fact, there is a stronger notion of uniform amenability for $\Gamma$ (...
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Trying to construct the ultrafilter 2-monad on $\mathbf{Cat}$
By which I mean, following Bôrger's paper Coproducts and Ultrafilters, the terminal monad among those that preserve finite coproducts, if such a thing can be constructed.
So far, what I have is, ...
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What is the "right" notion of exponentiation in $\beta \mathbb N$?
The Stone–Čech compactification $\beta \mathbb N$ of the positive integers has extensive applications in combinatorial number theory.
A feature of $\beta \mathbb N$ that makes these applications ...
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Non-existence of countable base of arbitrary ultrafilter
$\scr{B}$ is the base of a nonprincipal ultrafilter $\scr{U}$ on $\omega$ if 1. $\forall U,V\in\mathscr{B}~\exists T\in\mathscr{B}:~T\subset U\cap V$, 2. $\forall X\in\mathscr{U}~\exists U\in\mathscr{...
user137651
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Number of ultrafilters in an extender
Assume GCH. Suppose $j: V\to M$ is an elementary embedding such that $\mathrm{crit}(j)=\kappa$, ${}^\kappa M \subset M$, $M\supset V_{\kappa+2}$. We can assume $j$ is defined from some extender of ...
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Asymptotically discrete ultrafilters
Definition 1. A ultrafilter $\mathcal U$ on $\omega$ is called discrete (resp. nowhere dense) if for any injective map $f:\omega\to \mathbb R$ there is a set $U\in\mathcal U$ whose image $f(U)$ is a ...
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Relation between ultrafilters ${\scr U}$ and ${\scr U} \otimes {\scr U}$ [closed]
If ${\scr U}$ and ${\scr V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\scr U}\otimes{\scr V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\...
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Ultrapower of a field is purely transcendental
Let $F$ be a field, $I$ a set, and $U$ an ultrafilter on $I$. Is the ultrapower $\prod_U F$ a purely transcendental field extension of $F$?
According to Chapter VII, Exercise 3.6 from Barnes, Mack "...
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Dense subfilter of selective ultrafilter
Given selective ultrafilter $\mathcal{U}$ on $\omega$ and dense filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$, where $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ if the limit exists. Let $\...
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Models of $\mathsf{ZFC}$ with neither $P$- nor $Q$-points
A $P$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ there is $x\in {\scr U}$ such that the restriction $f|_x$ is either constant, or finite-to-one.
A $Q$...
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Dense filter and selective ultrafilter
We say that $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ is the density of subset $A\subset\omega$ if the limit exists. Let us define the filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$.
...
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Metrically Ramsey ultrafilters
On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...
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The property of the dense subfilter of a selective ultrafilter
Let us define the density of subset $A\subset\omega$ :
$$\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$$
if the limit exists. Let $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. $\mathcal{F_1}$ is the ...
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Some kind of idempotence of dense filter
In discussion of following questions question1, question2, question3 became clear (see definitions in question3 ) that for the Frechet filter $\mathcal{N}$ we have $\mathcal{N}\nsim\mathcal{N}\otimes\...
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Maximal elements in the Rudin-Keisler ordering
Let $\text{NPU}(\omega)$ be the set of non-principal [ultafilters][1] on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\...
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Non-tensor-representable ultrafilters on $\omega$
If ${\cal U}$ and ${\cal V}$ are ultrafilters on non-empty sets $A$ and $B$ respectively, then the tensor product ${\cal U}\otimes{\cal V}$ is the following ultrafilter on $A\times B$:
$$\big\{X\...
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Minimal cardinality of a filter base of a non-principal uniform ultrafilters
Let $\kappa$ be an infinite cardinal. An ultrafilter ${\cal U}$ on $\kappa$ is said to be uniform if $|R|=\kappa$ for all $R\in{\cal U}$. If ${\cal U}$ is a non-principal ultrafilter on $\kappa$, ...
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Ultrafilters as a double dual
Given a set $X$, let $\beta X$ denote the set of ultrafilters. The following theorems are known:
$X$ canonically embeds into $\beta X$ (by taking principal ultrafilters);
If $X$ is finite, then there ...
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The example of the idempotent filter or subsets family with finite intersections property
From the answer of Andreas Blass and comments of Ali Enayat on my question Selective ultrafilter and bijective mapping it became clear that for any free ultrafilter $\mathcal{U}$ we have $\mathcal{U}\...
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Nonexistence of a 'product universal' compact Hausdorff pseudotopological space?
I'm largely following the definitions of this paper, but I will replicate the relevant ones here.
I'm taking a pseudotopological space to be a set $X$ together with a relation $\rightarrow$ on the ...
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On infinite combinatorics of ultrafilters
Let $\mathcal{U}$ be (selective) ultrafilter and $\omega = \coprod_{i<\omega}S_i$ be partition of $\omega$ with small sets $S_i\notin\mathcal{U}$. All $S_i$ are infinite. Does there exist a system ...
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Selective ultrafilter and bijective mapping
For arbitrary (selective) ultrafilter $\mathcal{F}$ does there exist bijection $\phi:[\omega]^2\to\omega$ with the property: $\forall B\in\mathcal{F} : \phi([B]^2)\in\mathcal{F}$ ?
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A notion of largeness somewhere between $\mathrm{IP}_+$ and $\mathrm{IP}_+^*$
It is a well known fact that a set $A \subset \mathbb{N}$ is $\mathrm{IP}$ if and only if there exists an idempotent $p \in \beta \mathbb{N}$ (i.e., $p+p=p$) such that $A \in p$. Similarly, $B \subset ...
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Valuation Rings and Ultrafilters
I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter.
To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. ...
user30211
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Complexity of ultrafilter limits
Let $\mathscr{F}$ be a free ultrafilter on $\mathbf{N}$ and, for each $A\subseteq \mathbf{N}$ and $n \in \mathbf{N}$, define
$$
d_n(A):=\frac{|A\cap [1,n]|}{n}.
$$
Question. Considering $\mathcal{P}...
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Ultrafilters preserving infinite joins
A filter $U$ over a boolean algebra (isomorphic to a powerset) $A$ "preserves" a join $a = \bigcup_{i\in I}a_i$, if $a\in U$ implies $a_i\in U$ for some $i\in I$. (A join $a$ is infinite if $I$ is.) ...
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Completeness number of ultrafilters
In what I write below, by "ultrafilter" I mean a non-principal ultrafilter.
Given an ultrafilter $U$ on some set $S$, let $\mu$ be the least cardinal such that $U$ is $\mu$-complete but not $\mu^+$-...
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A property of an ultrafilter
Let $\mathcal U$ be a free ultrafilter on a set $X$. For $n\in\mathbb N$ let $\mathcal F$ be a family of $n$-element subsets of $X$ such that $\bigcup\mathcal F\in\mathcal U$.
Question. Is there a ...
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Are separability and ccc equivalent for closed subspaces of $\beta N$?
Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable ...
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Ideals on $\mathbb N$ and large sets that have small intersection
Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable:
$$A,...
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Identification of ultrafilters with measures
We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$.
Now my question is which ...
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Selectors for bases of ultrafilters
If $\mathcal{U}$ is a selective (Ramsey) ultrafilter on $\omega$ and $\mathcal{B}$ is its base, then for every sequence $A_0\supsetneq A_1\supsetneq A_2\supsetneq\ldots$ in $\mathcal{B}$ there exists $...
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Selective ultrafilter on $\omega$ is normal. Clear proof
In this question I had asked about proof of the property of selective ultrafilter. As was answered, the proof is trivial if we know that ultrafilter is selective iff it is Ramsey ultrafilter. The ...
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What are the components of the Stone-Cech Remainder?
Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a ...
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Set of subsequences with the same ultrafilter limit of the original sequence
Let $\mathscr{U}$ be a free ultrafilter on the positive integers $\mathbf{N}$ and fix $U \in \mathscr{U}$ such that $U$ is not cofinite (thanks J.D.Hamkins for the correction.)
Consider the natural ...
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Ultrafilters and diagonal arguments
Is there a diagonal argument to show that if $x$ is infinite then ${\cal P}(x)$ (the power set of $x$) is smaller than $\beta x$ (the set of ultrafilters on $x$)?
(Added later. I tried commenting ...
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Is there a 'local' version of Near Coherence of Filters?
The axiom Near Coherence of Filters (NCF) is known to be independent of ZFC.
Axiom (NCF I): For any two free ultrafilters $\mathcal D$ and $\mathcal E$ on $\mathbb N$, there exist finite-to-one ...
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Permutation on $\omega$ and Ramsey ultrafilter
let $\pi:\omega\to\omega$ be permutation and $\mathcal{F}$ is Ramsey selective ultrafilter on $\omega$. There are uncountable many increasing subsequences of $\pi$. Can one proof that one of them has ...
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Possible cofinalities of cuts of ultraproducts
Suppose $\kappa$ is a regular cardinal, $\bar{\mu}=(\mu_i: i<\kappa)$ is an increasing sequence of regular cardinals ($>\kappa$) and set
$pcut(\bar \mu)=\{ (\lambda_1, \lambda_2):$ for some ...
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The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder
If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
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"Gaps" in the Rudin-Keisler ordering
If $(P,\leq)$ is a poset and $p\in P$, then we say that $p$ is the lower part of a gap there is $q \in P$, $q>p$ such that $[p,q] = \{p,q\}$. (This is equivalent to the statement that $(\uparrow p) ...
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Infima in the Rudin-Keisler ordering
Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
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