Questions tagged [ultrafilters]

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6
votes
1answer
131 views

Regular limit points of possible cofinalities

Let $A$ be a non-empty set of regular cardinals such that $\vert A\vert <\text{min}\ A$, and $\{\nu_i\mid i<i_0\}\subseteq \text{pcf}\ A$ be a strict increasing sequence having limit length $i_0$...
2
votes
1answer
81 views

What is the smallest cardinality of a base of an ultrafilter on $\omega$ related to an almost disjoint family of cardinality $\mathfrak c$?

Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an almost disjoint family of infinite subsets of $\omega$. The almost disjointness of the family means that $A_\alpha\cap A_\beta$ is finite for any ordinals ...
0
votes
0answers
39 views

Notation of $P^+$-families - bibliography searching

have you ever met with notation of $P^+$-families in other papers than Iian B. Smythe "A local Ramsey theory for block sequences" and his phd? Thank you in advance
5
votes
1answer
181 views

NCF, P-points, weak P-points, and cardinalities

The post is a bit long, but all the questions are similar or concern the same topic. Let $\omega^*=\beta\omega\setminus\omega$. A well-known topological definition of a P-point (on $\omega$) is as ...
0
votes
1answer
150 views

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, ...
7
votes
1answer
226 views

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 ...
9
votes
1answer
314 views

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 ...
6
votes
1answer
227 views

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<\...
0
votes
0answers
102 views

Ultrafilter on $X^Y$

ZFC (+CH if necessary). Let $X,Y$ be infinite sets and $\cal{U}$ be free ultrafilter on $X^Y$. For any $Z\subset Y$ we denote $\pi_Z:~X^Y\to X^Z$ standard projection and $\pi_Z(\cal{U})$ the ...
3
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0answers
107 views

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 ...
7
votes
1answer
346 views

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 ...
3
votes
1answer
166 views

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 ...
5
votes
1answer
166 views

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?
3
votes
1answer
149 views

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 $\Gamma$ (...
1
vote
1answer
116 views

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, ...
5
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0answers
94 views

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 ...
1
vote
1answer
102 views

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{...
5
votes
1answer
287 views

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 ...
7
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0answers
150 views

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 ...
5
votes
1answer
191 views

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\...
4
votes
0answers
226 views

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 "...
2
votes
1answer
82 views

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 $\...
4
votes
1answer
287 views

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 $Q$...
3
votes
1answer
106 views

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 $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. ...
8
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0answers
209 views

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 ...
4
votes
1answer
173 views

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 ...
1
vote
1answer
105 views

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\...
0
votes
1answer
113 views

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 (\...
7
votes
1answer
147 views

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\...
3
votes
1answer
135 views

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 $\kappa$, ...
39
votes
2answers
2k views

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 ...
4
votes
1answer
97 views

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}\...
3
votes
1answer
149 views

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 ...
6
votes
1answer
272 views

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 ...
9
votes
1answer
261 views

Selective ultrafilter and bijective mapping

For arbitrary (selective) ultrafilter $\mathcal{F}$ does there exist bijection $\phi:[\omega]^2\to\omega$ with the property: $\forall B\in\mathcal{F} : \phi([B]^2)\in\mathcal{F}$ ?
3
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0answers
58 views

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 ...
6
votes
1answer
239 views

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 $K$. ...
4
votes
1answer
123 views

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}...
3
votes
1answer
102 views

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.) ...
12
votes
1answer
466 views

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 $\mu^+$-...
9
votes
3answers
448 views

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 ...
2
votes
1answer
97 views

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 ...
7
votes
5answers
436 views

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,...
3
votes
0answers
137 views

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 ...
4
votes
0answers
106 views

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 $...
3
votes
2answers
289 views

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 ...
3
votes
2answers
319 views

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 ...
4
votes
0answers
188 views

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 ...
22
votes
2answers
996 views

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 ...
8
votes
0answers
207 views

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 ...