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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Does the filter, $F$, on $S$ exist such that $p,q\in S$ and $p,q\in \lim{F}$

Consider two points $p,q\in S$ with $p\ne q$. Is it possible to find a filter, $F$, on $S$ such that all neighborhoods of $p$ and $q$ are contained in $F$? I would assume that when $p\ne q$ then, in ...
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3 votes
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Measurable cardinals, elementary embeddings, and Kunen's theorem

Suppose $\kappa$ is a measurable cardinal. Then if $U$ is the ultrafilter on $\kappa$, we can use this to generate an ultrapower of the entire universe. We can then embed $V$ into this ultrapower in ...
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1 vote
1 answer
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Find a sequence in a non-principal ultrafilter that decreases to a point

Let $\mathcal F$ be a non-principal ultrafilter of subsets of the set $X$. Let $x \in X$. Does there exist a decreasing sequence $F_1 \supset F_2 \supset ...$ in $\mathcal F$ such that $\bigcap_n F_n ...
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Showing that every convergent sequence in $\beta\mathbb{N}$ is eventually constant

I'm trying to show that every convergent sequence in $\beta\mathbb{N}$ is eventually constant. My professor told me to prove and use the following fact: Let $X$ be a Hausdorff space and let $(a_n)_n\...
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How can we prove that the ultrafilter lemma is not provable from ZF?

The ultrafilter lemma states that every proper filter on a set X is contained in some ultrafilter on X. Wikipedia says that in ZFC one can prove the ultrafilter lemma, but in ZF it's not possible to ...
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The filter generated by $\mathcal{B} \subseteq \mathcal{P}(I)$ is non-principal if and only if $\mathcal{B}$ has the finite intersection property

Would it be possible for someone to confirm whether the following proposition is erroneous. It appears in a set of lecture notes on model theory and pertains to non-principal filters. $\bf{Proposition}...
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principal ultrafilters proof

The following is a proof regarding principal ultrafilters from pg.2 of ULTRAFILTERS AND HOW TO USE THEM by Burak Kaya (https://users.metu.edu.tr/burakk/lecturenotes/village2019lecturenotes.pdf). ...
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Determining Initial state for Information Filter

Implementing a non-linear bearings-only information filter (Extended Information Filter?), the IF being the "inverse" of the classic Kalman Filter. I understand that the information matrix Y ...
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Is it possible to use a chunk of observations instead of one observation in Recursive Least Squares (RLS) at once?

Recursive Least Squares (RLS) by its structure reestimates coefficients iteratively utilizing one new observation in each iteration. Is it possible to use $n$ new observations in one iteration to ...
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Concise argument: $[f]$ in the ultrafilter that realizes $\Sigma$?

Let $\mathcal{L}$ be a countable language and $\mathcal{A}_i$ be $\mathcal{L}$-structures. Let $\mathcal{A}=\prod_{i \in \omega} \mathcal{A}_i/\mathcal{U}$, where $\mathcal{U}$ is a non-principal ...
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Order preserving map from interval $(0,1)$ to the ultrafilter

Let $\mathcal{A}= \prod_{n \in \mathbb{N}} \mathcal{A}_n /\mathcal{U}$, where $\mathcal{A}_n=(\{0, 1, \dots, n\},<)$ and $\mathcal{U}$ is a non-principal ultrafilter of $\mathbb{N}$. Does there ...
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Finding restriction of the ultraproduct that behaves like $\mathbb{Z}$

Let $\mathcal{A}= \prod_{n \in \mathbb{N}} \mathcal{A}_n /\mathcal{U}$, where $\mathcal{A}_n=(\{0, 1, \dots, n\},<)$ and $\mathcal{U}$ is a non-principal ultrafilter of $\mathbb{N}$. Can we find ...
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1 vote
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Diagonlization argument: show there is no countable basis for the filter of nbhs of 0

Prove also that there is no basis of nbhs of zero in this topology which is countable. My attempt: Idea: find a sequence of decreasing sequences (with positive elements) that converge to zero, and ...
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Characterisation of pretopological convergence

I'm reading Convergence foundations of topology - Dolecki and Mynard and I am somewhat surprised by their proposition V.1.1 which states that A convergence $\xi$ on a set $X$ is a pretopology if ...
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Normal $\kappa$-complete non-principal ultrafilter on measurable cardinal $\kappa$

Is it true that for every measurable cardinal $\kappa$ there is a normal, $\kappa$-complete, and non-principal ultrafilter on $\kappa$ ?
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Proof of Hausdorffness of sequentially Hausdorff space under its sequential topology

Under "Topology of sequentially open sets" section of the Wikipedia page Sequential Space, there is a claim which says any sequentially Hausdorff(i.e. every convergent sequence has a unique ...
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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 $\kappa$...
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Filters in Boolean Algebras

I started to study set theory; my question is about definition of filters in Cori-Lascar's book Mathematical Logic. Definition (Filter): A filter in a Boolean Algebra $\mathcal{A}=(A,+,\times,0,1)$ is ...
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1 vote
1 answer
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Boolean algebra of a certain size containing filter

In an attempt to adapt a proof that I found in a book to I different statement, I am wondering about the following question. Suppose $\kappa$ is strongly inaccessible. Then $ G= \lbrace \lbrace x \in \...
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Certain properties of uniform structure

I am trying to read about unifrom spaces from Introduction to Uniform Spaces, and I was wondering about some basic facts which I wasn't able to find there. Let $(X,\mathcal{E})$ be a uniform space. ...
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3 votes
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Extending a filter to an ultrafilter inside a coideal

Assume the axiom of choice, every coideal contains an ultrafilter (which is equivalently a minimal coideal), and every filter may be extended to an ultrafilter. Now suppose we have a coideal $\mathcal{...
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4 votes
1 answer
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Are all Hyperreal Infinitesimals representable by Monotonically Decreasing Sequences to 0?

I know there are many possible theoretical ways to built *R, including axiomatic and set-theoretic approaches. I am limiting my attention specifically to the Superstructure approach, perhaps best ...
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1 vote
1 answer
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How can I work with the net associated to a filter?

I have a question about the net which one can always associate to a filter. First let me write down our definition: If $\mathfrak{F}$ is a filter on $M$ then we define $$I_\mathfrak{F}=\{(A,p): A\in \...
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How do I find a filter respecting the following properties?

I'm sitting in front of the following exercise: Prove that if we let $\mathfrak{F}$ to be a filter on a topological space $X$ and $A\subseteq X$ with $A^c\notin \mathfrak{F}$ then there is a filter $\...
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How do I show this statement about convergence in filters?

I have the following problem: Let $\{(M_i,\tau_i)\}_{i\in I}$ be nonempty topological spaces where $I$ is arbitrary but non empty. Let $M=\prod_{i\in I} M_i$. Let $F$ be a filter on $M$ and denote by ...
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1 answer
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How do I prove the following statement about filters and associated nets?

I have the following problem: Let $\mathfrak{F}$ be a filter with associated net $(p_i)_{i\in I_\mathfrak{F}}$. Show that $p\in M$ is a cluster point of $\mathfrak{F}$ iff $p$ is a cluster point of ...
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Does $\mathbb{Q} \times $ Cantor set have a complete sequence of $\sigma$-discrete closed covers?

Does the space $\mathbb{Q} \times \mathcal{C}$ possess a complete sequence of $\sigma$-discrete closed covers? I am interested in this question, because if answered positively, the Theorem 1 in the ...
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On the proof of ultrafilter theorem.

I was self studying some model theory and found this article which suited my interests. Now, I completed reading upto the ultrafilter lemma in this article. Actually I didn't understand the proof ...
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Conventions for distinguishing the two senses of an ultrafilter on X

Are there any well-established conventions to distinguish the two senses of being an ultrafilter on a set $X$ (when $X$ happens to be equipped with an ordering)? This ambiguity is confusing at first; ...
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3 votes
1 answer
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Construction of ultrafilters in Heyting Algebras

Let $$f: H\rightarrow Z_{2}$$ be a Heyting Algebra homomorphism. Could anyone explain why $$f^{-1}(1)$$ is an ultrafilter? I know that it's a prime ideal. It's not clear to me because complementation ...
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how is cardinality of free ultrafilters to be understood

Let $X$ be an infinite set, $F$ the set of filters and $U$ the set of free ultrafilters on $X$. Then for the cardinalities we have $|U| \leq |F| \leq 2^{2^{|X|}}$. I have found several proofs that we ...
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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 ...
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5 votes
1 answer
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Is the characterization of Hausdorff spaces in terms of ultrafilter convergence equivalent to the ultrafilter lemma?

It can be easily proven using the ultrafilter lemma that if every ultrafilter on a topological space converges to at most one point, then the space is Hausdorff. My question is is whether this ...
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2 votes
0 answers
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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 ...
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2 votes
2 answers
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Every element in a non-principal filter is infinite

I have to prove that if $\mathcal{F}$ is a non-principal filter then $\forall F \in \mathcal{F}$, $F$ is an infinite set, I tried by the contrapositive and absurdum but couln't get any result using ...
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0 votes
1 answer
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How do I prove that a filter converges in a cartesian product?

I have the following problem: Let $\{(M_i, \mathfrak{M}_i)\}$ be nonempty topopological spaces. And let $M=\prod_{i\in I} M_i$. Let $\mathfrak{F}$ be a filter on $M$ and so $\mathfrak{F}_i=(\pi_i)_* \...
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Neighborhood base generated by uniformity

In Topology for analysis by A. Wilansky the theorem 11.1.2 states: Let $(X,\mathcal{U})$ be a uniform space, let $\mathcal{B}$ be [uniformity] base for $\mathcal{U}$ and let $x \in X$. Then $\{U(x) :...
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Does continuous imply pseudo-monotone?

Let $E$ and $F$ be real Banach spaces, brought into duality by $\langle f, e \rangle$ for $f\in F$ and $e\in E$. Let $E$ be equipped with a topology $\tau_E$ finer than the weak topology $\sigma(E, F)$...
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How can I prove this statement about filters?

I have the following question: Let $M$ be a finite set an $F$ a filter on $M$. Show that this filter has to be fixed My Idea was the following: Since $M$ is finite also $P(M)$ is. But since $F\...
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1 vote
1 answer
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Localization and Field [duplicate]

Let K be a field, I an infinite indexed set and A be the product ring $K^I$. For every $p\in Spec A$, prove that the localization $A_p$ is a field. In particular, $p \in Spm A$. This question was ...
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8 votes
2 answers
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Cardinality of ultraproducts

This might be a trivial question, but I couldn't see how to do this now. Given an infinite model $M$ and infinite cardinal $\kappa > |M|$ and a non-principal ultrafilter $F$ over $\kappa$, is it ...
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Lemma 2.20 of Kunen

I'm trying to understand the proof of a result in Kunen's Set Theory An Introduction to Independence Proofs Chapter VII, but I really can't find a way. We want to show that if $G$ is a $\mathbb{P}$-...
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filter design for two variables

I have two time varying input variables. I would like to design a filter on the multiplication of these two variables. I have two options if I choose 10 Hz as the cut off frequency: apply 10 Hz ...
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How to find filters in Boolean algebra $B=\{1,2,5,7,10,14,35,70\}?$

Find fitlers in the Boolean algebra $B = {1, 2, 5, 7, 10, 14, 35, 70}$ under the operations $+, \cdot, '$ defined by $$x + y = lcm(x, y)$$ $$x \cdot y = gcd(x, y)$$ $$x' = \frac{70}{x}.$$ I know that $...
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0 votes
1 answer
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How to fuse inertial and optical tracking data with an error-state kalman filter, and provide pose estimations for both update types?

I am searching for an error-state kalman filter that is able to fuse inertial and optical tracking data but provides pose estimates for both optical and inertial updates. Currently I am using the ...
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3 votes
1 answer
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ultrafilters as linear orders

In Henson's Model Theory lecture notes I found an exercise quite early on (1.30, p. 12) that prove too difficult for me. It goes like this: Let $L$ be the first order language whose only nonlogical ...
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Relation between a filter and its complement being an ideal.

Exercise. Show that a true filter $\mathcal{F}$ in $X$ is an ultrafilter iff the complement of $\mathcal{F}$ is an ideal, i.e., $Y,Z \notin \mathcal{F} \Leftrightarrow Y \cup Z \notin \mathcal{F}$, ...
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3 votes
1 answer
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Simple proof about prime filters.

Exercise. Let $F$ be an true filter in a set $X$. Show that $F$ is an ultrafilter ($\Leftrightarrow$ prime) iff a subset of $X$ intersects every set of $F$ then this same subset is in $F$. UPDATE. My ...
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5 votes
2 answers
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Ultrafilters and self-injections

In this answer, if I understand it right, the following theorem is stated: Let $X$ be a set and $f:X \to X$ an injective map. Let $\mathcal U$ be an ultrafilter on $X$ such that $f(U) \in \mathcal U$ ...
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1 vote
1 answer
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Prove a map $\varphi$ between all nets and all filters is surjective.

We define such a map: $\varphi(N) = \{A \subset X \ | \ \exists \alpha \in \Omega, \forall \beta \ge \alpha: f(\beta) \in A \}$ Here $N = (\Omega, f, \ge)$ is a net with index set $\Omega$. This way $\...
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