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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How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
Jakub Konieczny's user avatar
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Non-rigid ultrapowers in $\mathsf{ZFC}$

This question stems from a weakness in this recent answer of mine. Question: Can $\mathsf{ZFC}$ prove, without invoking first-order model theory, that for every countably infinite structure $\mathcal{...
Noah Schweber's user avatar
17 votes
1 answer
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What makes Fubini product (tensor product) of filters a natural operation?

In several texts you can encounter tensor product or Fubini product of two filters defined as follows: If $p$ is a filter on a set $X$, $q$ be a filter on a set $Y$, then $$p\otimes q=\{A\subseteq X\...
Martin Sleziak's user avatar
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Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO ...
Martin Sleziak's user avatar
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In ZFC, does every strucutre have an ultrapower that is universal?

I call a structure $M$ of cardinality $\kappa$ universal if it is $\kappa^+$-universal, i.e., there is an elementary embedding from any $N \equiv M$ to $M$ if $|N| \le \kappa$. (I think this is a ...
Pteromys's user avatar
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Dimension of An Ultraproduct Field as a Vector Space over Another Ultraproduct Field

Suppose $F \subseteq K$ are fields with $G$ an ultrafilter on an infinite set $X$. If $F^{\ast}$ and $K^{\ast}$ represent the ultraproducts respectively of $F$ and $K$, it is easy to see that $[K : F]...
D_S's user avatar
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Does there exist a $\lambda$-good, $\omega$-incomplete filter on $\kappa$ such that $P(\kappa)/\lambda$ is $\lambda$-saturated?

Let $\kappa$ and $\lambda$ be cardinals (we can assume $\lambda < 2^\kappa$). Does there exist a $\lambda$-good, $\omega$-incomplete filter on $\kappa$ such that the quotient $P(\kappa)/\lambda$, ...
Pteromys's user avatar
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Does one always need some choice to show the existence of nonprincipal ultrafilters?

The last sentence of the section "Types and existence of ultrafilters" of Wikipedia's article on ultrafilter (current revision) says: In ZF without the axiom of choice, it is possible that ...
Usagi's user avatar
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Limiting the size of near-coherence classes in $\omega^*$

We say that two ultrafilters in $\omega^*$ (i.e., two non-principal ultrafilters on $\omega$) are nearly coherent if there is a finite-to-one mapping $\varphi:\omega\to\omega$ such that $\beta\varphi$ ...
Forever Mozart's user avatar
5 votes
1 answer
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Ideal:Kernel :: Filter:?

I understand that the notion of a filter is in some sense dual to the notion of an ideal, at least in the context of Boolean algebras1. Let $f:{\mathbf A} \to {\mathbf B}$ be a Boolean algebra ...
kjo's user avatar
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Is there a robust version of the moving least squares or of the Savitzky–Golay filter?

Is there a name for the following type of filter? I want to filter a noisy signal $f(x) = f_0(x) + noise(x)$ (where $f_0$ is a noiseless signal), to get a filtered signal $f_\text{F}(x)$ while ...
HelloGoodbye's user avatar
4 votes
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Prime Ideal Theorem implies Hahn Banach Theorem

I am reading Jech's Axiom of Choice, and there is this exercise: chapter 2 Problem 19: Show that the Hahn-Banach Theorem follows from the Prime Ideal Theorem. I came up with a (possibly wrong) proof,...
mathlearner98's user avatar
4 votes
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Open Sets of Convergence Space Induce a Topology (and vice versa)

It's Christmas holidays and I have some time to read about filter convergence. But I am not sure about the following exercise: Let $X$ be a space equipped with a convergence (i.e. a relation $\xi\...
Syd's user avatar
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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 ...
Colin's user avatar
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What it the intuition behind basis, filters as neighborhoods and neighborhoods?

I know every definition, but as hard as I think I can't get the intuition behind such objects. For example, the definitions I am familiar with the most: Take $X$ a topological space with topology $\...
L.F. Cavenaghi's user avatar
4 votes
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Literature about Ultrafilters

I am in the early stages of planning my senior project and was wondering if anybody had some recommendations of literature about the applications of ultrafilters in social choice theory, along with ...
Marcus Dupree's user avatar
4 votes
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215 views

Need a fast algorithm of adaptive convolution

Good morrow, gentlemen! I have to apply some kind of adaptive filter to my function $f(x).$ I present each point of my signal as a Gaussian, whose bandwidth depends on its location (not the point of ...
Felix's user avatar
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History of Hindman's Theorem

At this blogpost about Hindman's Theorem, I read the following lines: 'I love the odd history so allow me to digress... etc. ' This sentence made me curious to know what this history looks like. ...
Suze's user avatar
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Tychonoff theorem (2/2)

In following I would like to post a proof of Tychonoff theorem using filters of closed sets. I would be grateful if you could find any mistakes in my proof (also if you read and don't find any ...
dolan's user avatar
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Can the reals be embedded in an ultraproduct of finite fields?

Is it possible (using ZFC or any other axiom system which is known to be as consistent as ZF) to prove the existence of an ultrafilter $\mathcal F$ on the set of prime numbers s.t. there is an ...
Roee Sinai's user avatar
3 votes
0 answers
84 views

Generalization of Łoś theorem to infinitary logic

"It is straightforward to check that, essentially by the same proof as for $\mathcal{L}_{\omega \omega}$, Łos’s Theorem 0.6 holds for $ \mathcal{L}_{\kappa \kappa } $ and ultraproducts by $\kappa$...
Matteo Casarosa's user avatar
3 votes
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152 views

Which group do you get by taking the ultraproduct on the finite symmetric groups (given a fixed non-principal ultrafilter U)?

I'm trying to get a better understanding of ultraproducts and their typical uses. To that end, I'm wondering what happens if we fix an ultrafilter $U \in 2^{2^{\mathbb{N}}} $ and look at $\prod_{i \in ...
Greg Nisbet's user avatar
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weak P-point that is not a P-point

Is there a fairly simple example of a compact Hausdorff space with a weak P-point (i.e. a point that is not in the closure of any countable set to which it does not belong) that is a not a P-point (a ...
user558840's user avatar
3 votes
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224 views

Why are filters called filters?

Given a set $S$, a filter $\mathcal{F} \subset \mathcal{P}(S)$ is a non-empty subset of the power set of $S$ that satisfies the follow conditions: $\emptyset \not\in \mathcal{F}$ Given $A$, $B \in \...
Lucas Giraldi's user avatar
3 votes
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55 views

When can centered sets be extended to filters?

Suppose $(\mathcal O, \leq)$ is an arbitrary poset. Let us say that $\mathcal O$ is compact if every $\mathcal C\subseteq\mathcal O$ which is centered (any finite subset of $\mathcal C$ has a lower ...
tomasz's user avatar
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Normal ideal over a singular cardinal

Let $\kappa$ a singular cardinal. I want to show that there is no normal ideal $I$ which contains all of bounded subsets of $\kappa$. If that ideal exists, I can to show that $club(\kappa)\subseteq I^...
YCB's user avatar
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why is a ring ideal not called a filter

A ring ideal can be characterized by the two rules: $$(a\in I) \wedge (a ~ \textrm{divides} ~ b) \implies b \in I$$ $$ a,b \in I \implies \textrm{gcd}(a,b) \in I$$ (the usual definition states $a,b \...
almaus's user avatar
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If every $\kappa$-complete filter can be extended to a $\aleph_1$-complete ultrafilter, then some $\lambda\le\kappa$ is measurable

I want to prove the following We have $\kappa$ an infinite regular cardinal such that every $\kappa$-complete filter on $\kappa$ can be extended to an $\omega_1$-complete ultrafilter on $\kappa$. ...
user avatar
3 votes
0 answers
284 views

Properties of ultrafilter limit of free ultra filter

Let $I$ be a directed set, $\mathcal U$ a free ultrafilter on $I$, i.e. an ultrafilter which contains the Fréchet filter (or equivalently $\bigcap \mathcal U = \emptyset$) and $(x_i)_{i\in I}$ a ...
Lukas Betz's user avatar
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Ideals and Filters are closed under Arbitrary intersection and Arbitrary union respectively

I just realised two things. $1)$ Ideals are closed under arbitrary intersection. $2)$ Filters are closed under arbitrary union. $X$ is a set.$I\subset \mathcal P(X)$ is called an Ideal if the ...
user118494's user avatar
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Filters, nets and Galois correspondence

In the lecture, our prof. mentioned that the correspondence between nets and filters is a Galois correspondence without giving any more details about that. In algebra, the proof of the Galois ...
mdot's user avatar
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Another question in relation to Tychonoff theorem

Let $X_i$ be compact topological spaces and let $X = \prod_{i \in I}X_i$ and let $\mathscr F$ be ultrafilter on $X$. Define $\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here $\...
dolan's user avatar
  • 537
2 votes
1 answer
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How to force character of ultrafilter be equal to $2^k$?

Let $k$ be an infinite cardinal. We already know there are exactly $2^{2^k}$ distinct non-principal ultrafilter on $k$. Here The set of ultrafilters on an infinite set is uncountable. And proof uses ...
Cezar's user avatar
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How many affine prime-quotient ultrafilters does a rational semiring have?

I know ultrafilters are considered powerful by more-learned mathematicians than I. I cannot profess to understand the reasons how and why although I can see the power of Zorn's Lemma and the axiom of ...
it's a hire car baby's user avatar
2 votes
0 answers
30 views

Extension of single zero-crossing property

Let $f\in\mathscr{C}^2(\mathbb{R},\mathbb{R})$ a strictly increasing function, striclty convex on $(-\infty,0)$, strictly concave on $(0,\infty)$ and let $\sigma_1>\sigma_2>0$ be two real ...
NancyBoy's user avatar
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Does this basic property of hyperreal function hold?

Let $x=(a_1, a_2, a_3, ...) + \mathcal U \in {}^\ast \mathbb R := \displaystyle\prod_1^\infty \mathbb R/\mathcal U$ be a hyperreal number using the ultrapower construction and $f \colon \mathbb R\to \...
Markus Klyver's user avatar
2 votes
0 answers
129 views

Is a filter generated by a Dedekind cut a minimal Cauchy filter and vice versa?

I'm reading The Reals as Rational Cauchy filter by I. Weiss and I am trying to establish a connection between Dedekind reals and Bourbaki reals. I'm working with this definition of a Dedekind real a.k....
A Dz's user avatar
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60 views

Are ultrafilters precisely the designated sets in semantics(es) for classical logic?

I'm messing around with ultrafilters to try to understand them better. It seems as if ultrafilters work well as sets of designated truth values in matrix semanticses for classical logic. This also ...
Greg Nisbet's user avatar
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Proving that a family of sets is a free filter

We say that a subset $A$ of $\mathbb{N}$ has density if the limit $$\lim_{n\to\infty} \frac{|A \cap [n]|}{n}$$ exists (where $[n] = \{1,2,\dots,n\}$). Prove that the family $\mathcal{F}$ of subsets ...
Giorgos Giapitzakis's user avatar
2 votes
0 answers
73 views

Are Banach-Mazur Games related to filter Convergence?

Is there a way to connect filter convergence to the condition for player 1/2 to win a Banach-Mazur game in an if (and only if) fashion? Thanks! Details below... A Banach-Mazur game is defined as ...
Zach466920's user avatar
  • 8,341
2 votes
0 answers
44 views

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 ...
TreshUp's user avatar
  • 21
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0 answers
54 views

A filter in a subset of a topological vector space by a sequence

Let $E$ be a topological vector space, $A \subseteq E$ and $S:=\{x_1,x_2,x_3,\cdots\}$ a sequence in $A$. I know that, if for each $n\in \mathbb{N}$, we define $$S_n:=\{x_{n+1},x_{n+2},\cdots\}$$ then ...
Guilherme's user avatar
  • 1,591
2 votes
0 answers
117 views

Subset of a unit interval generated by a non-principal ultrafilter is non-measurable

I was reading about use of ultrafilters in general topology and found a document by Dror Bar-Natan. It contained this problem, so I embarked on solving it. In my opinion, it's rather interesting. Let ...
Nik Bren's user avatar
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2 votes
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Is ultrafilter convolution a special case of convolution of probability measures?

If $G$ is a semigroup and $\beta G$ is the set of ultrafilters on $G$, then $\beta G$ is also a semigroup: given $p,q\in \beta G$, we define $$p*q := \{E\subseteq G : \{g\in G : g^{-1}A\in q\}\in p\}.$...
Ehsaan's user avatar
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2 votes
0 answers
43 views

Prove that $\{x\in X\mid x\cap\kappa$ is $<\gamma$-closed$\}\in\mathcal I^*$.

Suppose that $\mathcal I$ is a $\kappa$-dense and normal ideal in $X$ (that is $\mathcal P(X)/\mathcal I$ has a dense set of power $\leq\kappa$), where $\kappa\subset X$ and $\kappa>\omega$ is ...
edgar alonso's user avatar
2 votes
0 answers
24 views

Is it possible to cluster particles and then resample each cluster?

Is it possible to cluster particles (in the particle filter) and then resample each cluster separately? if yes, the resampling are done parallel?
nil's user avatar
  • 21
2 votes
1 answer
336 views

Is every filter generated by a (non-trivial) filter base?

Suppose $F$ is any filter defined on a set $X$ (that is, with the partial order given by $\subset$). I know that every filter base generates a filter and that a filter is always a filter base itself (...
R Los's user avatar
  • 756
2 votes
0 answers
103 views

Proving that a set is an ultrafilter.

A sequence of ultrafilters of $\omega$ (therefore each $p_n \in \beta \omega$) $p_n$ is discrete if there exists a sequence $X_n$ of subsets of $\omega$ such that for each $n$, $X_n \in p_n$ and if $n ...
user avatar
2 votes
2 answers
178 views

Injective map preserves ultrafilter (?) Proof

This is a theorem in Willard's text, Theorem 12.14, p81, which states: $f$ maps $X$ into $Y$ and $\mathcal{F}$ is an ultrafilter on $X$, then $f(\mathcal{F})$ is an ultrafilter on $Y$. We define $f(\...
Bryan Shih's user avatar
  • 9,518
2 votes
0 answers
184 views

History of the theory on Ideal convergence

I have mentioned in my recent posts that I'm studying Ideal-convergence. So the thing is I am expected to make a small presentation on this topic in two weeks on what I have learnt so far and ...
user118494's user avatar
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