Questions tagged [filters]

Filters (and ultrafilters) are used in various areas of mathematics, e.g. general topology, set theory, boolean algebras, model theory. In topology they can be used to study convergence in a more general way than just convergence of sequences. For questions about filters in the sense of signal processing, do not use this tag and instead use (signal-processing) or another appropriate tag.

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smallest filter

Below in the block quotes is what I am trying to prove. Below the block quotes is my attempt for the first part of part (b). I guess I don't see what makes part a and part b different. For part a ...
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Another Topology on the Prime Spectrum of a Ring

Let's fix a commutative ring $R$. We'll write $X = \operatorname{Spec}R$ for the set of prime ideals of $R$. Finally, let's write $\beta X$ for the Stone-Čech compactification of $X$. We can define a ...
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Why care about ultrafilters (in operator algebras)?

I recently learned about filters and ultrafilters and the notion of convergence with respect to an ultrafilter in a topological space. I have to say, at first I could not appreciate their potential. I ...
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Principal Ultrafilter contains a finite set

Prove that an ultrafilter is principal if and only if it contains a finite set. I can't quite figure out why and it may be because I don't have a clear understanding of a principal ultrafilter. I have ...
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2answers
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Images of filters need not be a filter

My question is the following: Let $f$ be a mapping of a set $X$ into a set $Y$ and $\mathcal{F}$ a filter on $X$. Why is it that $f\left(\mathcal{F}\right)=\{f[F]:F\in\mathcal{F}\}$ is not a filter ...
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Anistrophic Deriche filter

I'm currently studying a Deriche filter. I've managed to create python program for derivative smoothing from this article [1]. As I understand it, Deriche used one $\alpha$ for smoothing in both ...
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1answer
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How many nonprincipal ultrafilters exists on $X$ such that $|X|^2=|X|$?

Let $X$ be an infinite set, idemmultiple ($|X|^2 = |X|$) if that helps. I am looking for a proof that $X$ has more than $|X|$-many ultrafilters. It has $|X|$-many principal ultrafilters. Without at ...
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A net clusters at $x$ iff the “filter generated by the net” clusters at $x$

Let $\langle x_\lambda \rangle_{\lambda\in\Lambda}$ be a net. Let $\mathscr{F}_{x_\lambda}$ be the filter generated by the net $\langle x_\lambda \rangle$, i.e. the generated by the filter base $\...
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1answer
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What is the limit $x\to\infty$?

I'm working on some exercises about bases of filters and in one of them they want me to compare the bases $x\to+\infty, x\to-\infty$ and $x\to\infty$. I know that I could take the base $\{(x,+\infty):...
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3answers
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Convergence of a filter defined in a subspace

I am reading about convergence of filters in topological spaces and cluster points. My doubt is the following: let $X$ be a topological space, $A\subset X$ and $\mathcal F$ a filter on $A$. Given $x \...
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Filters in Topology (Converge)

Let $X$ be an infinite set, $\mathscr{F}$ the filter on $X$ generated by the filter base consisting of all complements of finite sets. To which points does $\mathscr{F}$ converge if $X$ is given the ...
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Convergence of filters in top.

Let $X$ be any set equip with the dis. topology and $x\in X$. What filters $\mathscr{F}$ converge to $x$, written $\mathscr{F}\to x$? A filter $\mathscr{F}$ is said to converge to $x$, written $\...
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Consider a net and its associated filter, then the associated net of the filter is the original net?

Take a net $s: D \rightarrow (X,T)$ and then its associated filter, $\mathcal{F}_s$. Then we take the associated net of this filter, $s_{\mathcal{F}_s}$. Do $s$ and $s_{\mathcal{F}_s}$ coincide? It ...
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Non-principal Ultrafilter on Measurable Subsets of $\mathbb R$

Let $\Omega$ be the set of all measurable subsets of $\mathbb R$ ordered by inclusion. Is it possible to construct a non-principal ultrafilter on $\Omega$?
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1answer
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Ultralimits vs limits of subsequences

Let $(x_n)_{n \geq 1} \subset \mathbb{R}$ be a sequence of real numbers, and let $\omega \subset \mathcal{P}(\mathbb{N})$ be a non-principal ultrafilter. We say that $x$ is an ultralimit of $(x_n)_{n \...
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“Proof” that every ultrafilter has a least element

Ultrafilters come in the principal and the free variant. Elsewhere it is said that principal is equivalent to the ultrafilter having a least element, i.e. one that is contained in every other element ...
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Thin sets in free ultrafilters

Let's all a set $A\subseteq \omega$ (where $\omega$ denotes the set of non-negative integers) thin if $$\lim\sup_{n\to\infty}\frac{|A\cap\{0,\ldots,n\}|}{n+1}=0.$$ Is it true that there are free ...
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Ultrafilters on naturals determine p-adic numbers

While preparing for a short lecture on ultrafilters for undergraduates, I realized some interesting things I have never read about. Though I'm asking now a specific question, any reference about this ...
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Can a set have more than one ultrafilter?

https://proofwiki.org/wiki/Definition:Ultrafilter_on_Set Given the definition that a filter $\mathcal F $ on a set $\mathcal S$ is an ultrafilter on $\mathcal S$ iff there is no filter on $\mathcal ...
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1answer
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A union of proper filters in a chain is a proper filter

I was looking into the proof of the Ultrafilter lemma by Zorn's lemma, and I wasn't able to find a sufficient explanation for the following claim: If $\mathcal{C}\subseteq P$ is a chain of proper ...
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2answers
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Filter and ultrafilter properties in Boolean algebra

In a Boolean algebra $(\mathbb{B}, \vee, \wedge, 0, 1, \neg)$, a ultrafilter $U \subseteq \mathbb{B}$ satisfies If $a \wedge b \in U \iff a \in U$ and $b \in U$. If $a \vee b \in U \iff a \in U$ or $...
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1answer
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Boolean algebra's filter as a partially order's filter

This is a basic question I'm trying to figure out: why the Boolean's filter definition corresponds to the order-theoretic definition of filter ? Here follows the relevant definitions. Definition 1 (...
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1answer
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Let X be a non-empty set and U an ultrafilter over X…

Let X be a non-empty set and U an ultrafilter over X. Show that if X = $A_1 \cup A_2 \cup… \cup A_n$, then there is k $\in$ {1,…, n} such that $A_K \in$ U. X = $A_1 \cup A_2 \cup… \cup A_n$ = $\...
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Let X and Y be non-empty sets, f: X $ \to $ Y a function and F a filter on X…

Let X and Y be non-empty sets, f: X $ \to $ Y a function and F a filter on X. Show that $f_*$ (U) = {V $ \subset $ Y: $ f^{- 1} $ [V] $ \in $ U} is a filter on Y. Also show that if F is an ultrafilter ...
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1answer
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Is distributivity of a lattice needed if we want its collection of prime filters to be a Stone space?

Let $L$ be a bounded distributive lattice and let $PF(L)$ denote its set of prime filters. It is well known that $PF(L)$ is a Stone space if it is equipped with the topology that has sets of the form $...
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1answer
44 views

Translation invariant ultrafilters?

Is there an ultrafilter $\mathcal{U}$ on $\mathbb{N}$ such that $\mathcal{U}-n = \mathcal{U}$ for all $n \in \mathbb{N}$? Here, $$\mathcal{U}-n = \{ A - n: A \in \mathcal{U} \}$$ and $$A - n = \{ m \...
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1answer
51 views

Ultraproduct of parwise non-equivalent structures

It is easy to see that the ultraproduct of a family of structures over a principal ultrafilter is elementarily equivalent to the structure whose index generates the ultrafilter, i.e. if $U$ is ...
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Are topological spaces (and open maps) comonadic over sets (and functions)?

In the neighborhood formulation of topological spaces, the data of a topological space is a pair $(X,N)$ of a set $X$ and a function $N : X \to F X$, where $FX$ denotes the set of filters of subsets ...
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1answer
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Prove via ultraproducts that the class of algebraic extensions is not axiomatisable in the language of rings

My idea was to take the ultraproduct of $Q_n = \mathbb{Q}(\sqrt[n]{2})$ over some non-principal ultrafilter and then let $T_n$ be the (first-order) sentence "There are at least $n$ linearly ...
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Given a signal I(m,n) = a + b*m + c*n and a non linear filter e.g. median filter, find the response.

I stumbled upon this question and I can't seem to solve it correctly "Given a signal $x(n,m) = a +b{\cdot}m + c{\cdot}n $ and a filter, find the processing system response defined in a finite ...
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1answer
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limit along ultrafilter is multiplicative

I am trying to prove that the limit of a sequence along a non-principal ultrafilter on $\mathbb{N}$ is multiplicative and I am getting stuck. This is what I have until now. Let $U$ be a non-principal ...
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25 views

Existence of ultrafilter on the positive integers

Let $(a_{n,k}: n,k \ge 1)$ be an infinite matrix of positive reals such that $\sum_{k\ge 1}a_{n,k}=1$ for all $n\in \mathbf{N}$. Question. Is it true that there exist a free ultrafilter $\mathscr{F}$ ...
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1answer
146 views

Linear iterability for set models of $\mathsf{ZFC}$

As I found some problems in my writing and proof, I have heavily edited my questions. So I am reading this corollary in Steel's notes, found here, and I am confusing myself over and over again. I ...
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0answers
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Ultraproduct construction: are finite hyperreals just a thinly disguised version of Cauchy sequences?

Periodically I've tried to wrap my head around nonstandard calculus and hyperreals, but I always thought I needed a lot more of a background in formal logic and/or set theory to understand what's ...
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1answer
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If countable fragments of an ultrapower can be realized, then the ultrafilter is $\omega_1$-complete

This is exercise 9 of Steel's notes on ultrapowers, found here. I have managed to progress through it, but I am stuck and I would appreciate any help. So this is the statement: Let $M\models \mathsf{...
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2answers
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What happens to the Stone-Cech compactification if you change “compact Hausdorff” to “T1 compact”?

I have asked a similar question elsewhere: what happens to the Stone-Cech compactification if you change “compact Hausdorff” to “$T_1$ compact”? Here, I've added $T_1$ as opposed to this. Is $K$ there ...
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1answer
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Skolem hulls in arbitrary models of some fragments of ZFC

So I am trying to learn a little bit about iteration trees, and have decided to read a note by Steel, found here. On page 2, just after exercise 3, he uses the Hull "operator" in a way which ...
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Are there filters other than principal filters? If yes, when?

Consider a Boolean algebra $\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$. Let $\mathcal{F}_\mathcal{B}$ denote the set of all filters on $\mathcal{B}$, and let $\mathcal{F'}_\mathcal{B}$ denote the set of ...
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What happens to the Stone-Cech compactification if you change “compact Hausdorff” to “compact”?

Here if in the diagram below Universal property and functoriality we take a modification that any continuous map $f:X\to K$ where $K$ is compact but not necessarily Hausdorff what would be our $\beta ...
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1answer
50 views

Topology of $P(\omega)$ as $2^\omega$ (ultrafilters are open?)

I'm looking for a sanity check more than anything else. We may identify $P(\omega)$ with $2^{\omega}$ in the obvious manner. Let $U$ be a principal ultrafilter on $\omega$, so $U \subseteq 2^{\omega}$ ...
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Show that atom of Boolean algebra B inside B' is also the atom of B' where B' is subalgebra of B. [closed]

Let B := (B, ≤, ∨, ∧, c , 0, 1) be a Boolean algebra and B' := (B' , ≤, ∨, ∧, c , 0, 1) be a Boolean subalgebra of B. Show that an element of B0 that is an atom of B must also be an atom of B' . ...
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Is $\mathcal{F}$ (an ultrafilter on $\mathbb{N}$ such that $\{1, 3, 5\} \in \mathcal{F}$) a non-principle/free filter?

Let $\mathcal{F}$ be an ultrafilter on $\mathbb{N} = \{0, 1, 2, 3, \dotsc\}$ such that $\{1, 3, 5\} \in \mathcal{F}$. Can we find out whether $\mathcal{F}$ is a principle or non-principle filter? I ...
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1answer
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$X \subseteq \mathcal{P}(B)$ contains filters on $\mathcal{B}$, are $\bigcap_{F\in X}F$ and $\bigcup_{F\in X}F$ filters too?

Consider a Boolean algebra $\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$ and $\phi \neq X \subseteq \mathcal{P}(B)$ whose elements are filters on $\mathcal{B}$. Show that: $\bigcap_{F\in X}F$ is also a ...
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1answer
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Relation between $h:B\to \{0,1\}$ and prime filter $F \subseteq B$

$\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$ is a Boolean algebra. Prove that: For any prime filter $F$, there is a homomorphism $h:B\to\{0,1\}$ such that $F = \{x\in B:h(x)=1\}$ For a homomorphism $h:B\...
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1answer
19 views

What is the smallest filter in $\mathcal{B}$ containing $a \in B$?

$\mathcal{B}:=(B,\leq,\lor,\land,^c,0,1)$ is a Boolean algebra. Show that the smallest filter in $\mathcal{B}$ containing $a\in B$ is given by $$F(a) = \{x\in B:a\leq x\}$$ Here's my work: First, I ...
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1answer
29 views

How to show that if x is a cluster point of a filter then it a cluster point of each of its associated or derived net.

Am stuck with this problem. However I am able to show the other way round. Please help me wih it.
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1answer
46 views

Obtaining linear functionals on $B(H)$ using ultrafilters.

Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i}\in \mathbb N$ and $X=H^*\otimes^\pi H$ is the projective tensor product. We have a natural isometry $$J:X\to X^{**}=B(H)^*$$ given by ...
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1answer
21 views

Show that any compact uniform space (compact for the topology of the uniformity) is complete

My efforts: Let the uniform space be $(S,\mathcal{U})$. For a Cauchy net {$x_\alpha$}, the collection of all $B_\gamma$ = {$x_\alpha:\alpha\geq\gamma$}, $\gamma\in I$, is a filter base that extends to ...
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1answer
36 views

Existence of ultrafilter and real sequence

Let $(a_{n,k}: n,k \ge 1)$ be an infinite matrix of positive reals such that $\sum_{k\ge 1}a_{n,k}=1$ for all $n$. Question. Is it true that there exist a free ultrafilter $\mathscr{F}$ on $\mathbf{N}...
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1answer
20 views

evaluating gives identity on sets from the basic model

I have a very basic question about forcing and generic set: why for $a\in M$ from the ground model do we have $$\dot{a}[G]=a$$ ? Where it is used in this equality here that $a\in M$ ?

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