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.
929
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How can I prove the existence of delta-incomplete / countable incomplete ultrafilters?
I am reading a W.A.J Luxembourg paper about nonstandard analysis (https://www.jstor.org/stable/3038221). He presents the following definition
I am stuck trying to prove the existence of delta-...
- 21
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0
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106
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Equivalent statements of "weak" choice
We can state the axiom of choice ($\mathsf{AC}$) in the following way $$\forall x\Big(\neg(\varnothing\in x)\longrightarrow\exists f\big(f\text{ is a function from $x$ to $\bigcup x$ }\wedge \ \forall ...
- 394
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303
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Does the term "free" in "free ultrafilter" have a meaning related to category theory?
I know that free ultrafilters are defined in contrast to principal/fixed ultrafilters. Nonetheless, is there some categorical way to view the use of the word "free" here (e.g. some pair of ...
- 243
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1
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78
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How to show this property of ultrafilter on a set $X$
Def:
Filter: A filter on a set $X$ is a collection $F \subset P(X)$ such that satisfies $a,b,c$
Filter basis: A filter on a set $X$ is a collection $F \subset P(X)$ such that satisfies $a,b$
...
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1
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1
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39
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Properties of proper maps using filters
I am reading a book on covering maps in Bourbaki-style with the following definitions:
A map $f:X\to Y$ is separated if for every $x,x'$ in the same fiber, there exist open disjoint neighbourhoods of $...
- 2,350
2
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0
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48
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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 ...
- 103
2
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1
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107
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Ultrafilters arising from homomorphisms
Reading a paper from Bagaria and Magidor I stumbled upon a theorem which's probably well-known (but not to me at the time), the Łoś-Eda theorem. In the form stated:
There exists a nontrivial group ...
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9
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1
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148
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Is every "filter" of rings principal?
Let $\mathcal{F}$ be a class of rings* with the following property:
If there is a homomorphism of rings $A \to B$ with $A \in \mathcal{F}$, then $B \in \mathcal{F}$. Moreover, $\mathcal{F}$ is closed ...
- 159k
3
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1
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59
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Intersection of a filter of clopen sets is of pozitive measure?
Let $X$ nonempty set and $\tau$ a compact topology on it. Let $\mu$ be a measure on the Borel $\sigma $- algebra of $\tau$.
Suppose we have a filter $\mathcal{F}$ of clopen sets such that $\mu(f)\geq0....
- 103
2
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2
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61
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Can an Ultrafilter be defined in terms of the Convergence of a Sequence?
We know that the set $\{\sin(0), \sin(1), \sin(2), ... \}$ is dense in the interval $(-1,1)$.
So now consider the sequence $S$ = $\langle \sin(0), \sin(1), \sin(2), \ldots \rangle$. This sequence ...
- 370
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0
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87
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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 ...
- 541
1
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32
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Explicit example of a discontinuous function between convergence spaces with the preimage of every open set open
The definition of continuity for convergence spaces is in terms of filters, but it’s still true that the preimage of any open set is open. Since this is no longer taken as the def of continuity, I ...
0
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1
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23
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Small doubt concerning treatment of filters in v. Dalen's L&S
This is a rather miniscule —and most likely borne of misunderstanding— doubt concerning the treatment and definition of "filters" (in particular their characterization as "proper") ...
- 152
1
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0
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42
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Is this condition redundant (neighborhood filter for TVS in Trèves)?
In Trèves's Topological Vector Spaces, Distributions and Kernels, Theorem 3.1 is as follows.
A filter $\mathscr F$ on a vector space $E$ is the filter of neighborhoods of the origin in a topology ...
- 6,483
3
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1
answer
223
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A property related to cofinal similarity
A partial order $\le$ is reflexive, transitive and antisymmetric. It is directed, if for all $x, y$ in the set there is $z$ such that $x \le z$ and $y \le z$.
Let $X, Y$ be partially ordered, directed ...
- 3,132
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0
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40
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Is every free ultrafilter on the natural numbers identical modulo permutation of the numbers? [duplicate]
Question
Is every free ultrafilter on the natural numbers identical modulo permutation of the numbers?
My thoughts
I just thought this was interesting because knowing how much structure all the free ...
- 9,151
0
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1
answer
37
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If a function preserves all filter limit points it is continuous
In Pete L. Clark's notes on convergence it is stated that (Proposition 5.14):
For a set function $f:X \to Y$, if for every prefilter $F$ on $X$ with a limit point $x$, $f(F)$ has $f(x)$ as a limit ...
- 2,293
0
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1
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140
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What does it mean to explicity exhibit something (modulo a proof of its existence) which cannot be explicitly exhibited?
I was reading this answer that no free ultrafilter can be exhibited on the natural numbers.
I have as a theorem that if the Collatz conjecture is true then the following is a free ultrafilter on the ...
- 9,151
2
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0
answers
72
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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 ...
- 9,151
3
votes
2
answers
64
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Quotient of infinite product of fields which is semi-artinian
Let $I$ be an infinite set (e.g. $I=\mathbb{N}$) and let $k$ be a field. The ring $R=\prod_I k$ is a notorious example of a ring which is absolutely flat (alternatively said von Neumann regular) and ...
- 2,015
2
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1
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53
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A bounded increasing net converges – formulated using filters
We know the following monotone convergence theorem:
If $n:X\to\mathbb R$ is a bounded increasing net, then $n$ converges. (Also for other spaces than $\mathbb R$.)
I am trying to formulate this ...
- 208
0
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0
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17
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No filter generated by a described family with the SFIP can be ultrafilter
Let $\mathcal{F} \subseteq \mathcal{P}(\mathbb{N})$ a "described", countable family of subset of $\mathbb{N}$ with the strong finite intersection property (that is, every nonempty finite ...
- 277
2
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0
answers
29
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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 ...
- 305
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62
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Prove that if $\mathcal{F}[x_d]\subseteq \mathcal{G}$, then there's a subnet $(y_e)$ of $(x_d)$ such that $\mathcal{G}=\mathcal{F}[y_e]$.
Let $X$ be a set.
Given any net $(x_d)_{d\in D}$ in $X$, we define
$$\color{red}{\mathcal{F}[x_d]}:=\big\{F\in 2^X:(\exists d_0\in D)(\forall d\in D)\big(d_0\preceq _Dd\Rightarrow x_d\in F\big)\big\}.$...
- 909
0
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0
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15
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Why is there an asymmetry in the scale of these filter kernels?
In this video, the kernel for a low pass filter is given to be $[\frac{1}{2},\ \frac{1}{2}]$, while the kernel for a high pass filter is given to be $[-1,\ 1]$.
Interpreting these as essentially ...
- 1,674
2
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1
answer
84
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Measurable Cardinals, Elementary Embeddings, and building up to transitive collapse
I am looking at the proof that $\kappa$ a cardinal is measurable iff it is the critical point of an elementary embedding from $V$ to an inner model $M$, specifically the forward direction.
I ...
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1
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1
answer
46
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Volatility as an Envelope for a Time Series
In time-series analysis, I keep running into graphs that look roughly like...
Recalling a trig computation from an ODE book which explored the acoustic phenomena of beats, I noticed the orange ...
- 1,674
6
votes
1
answer
82
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What length recursion constructs a free ultrafilter on $\Bbb{N}$?
Assume ZF together with some choice principle needed to make the choice in the following recursion that constructs a free ultrafilter on $\Bbb{N}$
$\mathcal{F}_0$ is the filter of co-finite subsets ...
- 2,699
0
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1
answer
41
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Sufficiency for existence of $lim_{\mathcal{B}}f(x)$, Zorich
I am working on Mathematical analysis 1 by Zorich and i am stuck at the proof for the sufficiency of the cauchy criterion for the existence of a limit of a function (page 132)
I am understanding all ...
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2
votes
1
answer
80
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If the ultrafilter space is Hausdorff, must the base space be discrete?
The question is in the title. Most of this post is a contextual preamble and some of my thoughts on the matter, I have not made much solid progress (and don't even know if this is true...)
$\...
- 35k
2
votes
2
answers
121
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Different definitions of filters in topology
So I have come across three different definitions for set theoretic filters that are used in topology, all of which only vary in their first axiom. The three different first axioms for a filter over a ...
- 357
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0
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38
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free ultrafilter question
I asked a question here Is this a free ultrafilter, and I'm attempting to write up the solution.
$\mathbb{N}=\{1,2,...\}$ and for any $A\subseteq\mathbb{N}$, $A+1=\{a+1:a\in A\}$.
Suppose $U$ is a ...
- 3,001
0
votes
1
answer
54
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Is this a free ultrafilter
Suppose $U$ is a free ultrafilter on $\mathbb{N}$. I'm trying to understand what is the ultrafilter generated by the collection $$G=\{E\subseteq\mathbb{N}: E=A+1, \ for \ some \ A\in U\}$$. I think ...
- 3,001
3
votes
2
answers
140
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What is the relationship between ultrafilters and propositional theories?
I am learning more about how to use ultrafilters by using them to prove several of the typical results which appear as applications of propositional (or first-order) compactness.
Generally speaking, ...
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1
vote
1
answer
78
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A question on existence of ultrafilters and cardinality.
Let $S$ be an infinite set with cardinality $k$. Let $U$ be an ultrafilter containing the filter of cofinite sets.
Then, for any set $P \subset U$ such that the cardinal of $P$ is $<k$ , is it true ...
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1
answer
87
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Dual space of $\mathcal{P}(M)$, when regarded as an $\mathbb{F}_2$-vector space.
This question stems from the well-known observation that the power set $\mathcal{P}(M)$ of any set $M$ can be given an $\mathbb{F}_2$-vector space structure by the symmetric difference operation.
Any ...
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5
votes
1
answer
173
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Ultrafilter proof of the infinite Ramsey Theorem
The Infinite Ramsey Theorem ($\mathsf{RT}$) is the statement
For any $r,p\in\mathbb{N}^{+}$ and any infinite set $A$, any $r$-coloring $c$ of $[A]^{p}$ has an infinite homogeneous subset.
Where $[r] ...
- 4,078
2
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0
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62
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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 \...
- 1,235
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votes
1
answer
61
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The complement of a filter is an ideal
In the book I'm reading, the author defines a filter like this:
Given a non-empty set $X$ a filter over $X$ is a set $f \subseteq P(X)$ such that:
(i) $f \not= \emptyset$
(ii) If $S_{1}, S_{2} \in f$ ...
- 575
1
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2
answers
78
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Examples of filters over the natural numbers
Given a set $X$, we can define a filter $F \subseteq 2^{X}$ (here $2^{X}$ means the powerset of $X$) as a family of sets that have the following properties:
(i) $F \not= \emptyset$
(ii) If $S_{1}, S_{...
- 575
0
votes
1
answer
31
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A two valued measure forms an ultrafilter
I've got stuck in verifying that for a two valued measure $\mu$ this forms an ultrafilter:
$$U=\{X\subseteq S:\mu(X)=1\}$$
E.g. why if $X\in U$ and $Y\in U$ then $X\cap Y\in U$ ?
- 3,872
4
votes
2
answers
100
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Proof for an ultrafilter [closed]
The question is as follows: Let $A,B$ be sets and $f: A \to B$ be a function. Let $U$ be an ultrafilter on the set $A$. Prove that the set ${\{C \subseteq B : f^{-1}C \in U}\}$ is an ultrafilter on $B$...
- 41
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0
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20
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Second order filter
Can I express this summation expression with the following routine logic:
$$\hat{x_{k+2}} = \sum_{i=1}^{k+2}\frac{x_i}{k+2} = \sum_{i=1}^k \frac{x_i+x_{k+2}+x_{k+1}}{k+2}$$
now suppose I want to ...
- 23
-1
votes
1
answer
69
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When the identity is the least non-constant
In the Handbook of Set Theory, what is the order on the page $25$ -$3$rd line $\leq$
and how can I see that $(\text{id}_U)$ is in the domain of $\pi$ ?
Here $\pi$ is a collapsing isomorphism of the ...
- 3,872
3
votes
1
answer
143
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Ultrafilters and compactness
A topological space is compact if and only if every ultrafilter is convergent.
While I was reading the proof of the one Side of theorem above, there is something I could not understand. Following is ...
- 155
0
votes
0
answers
25
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Is this double exponential smoothing algorithm valid?
I use this double smoothed prediction algorithm:
https://cs.brown.edu/people/jlaviola/pubs/kfvsexp_final_laviola.pdf
Equations:
$$
First Smoothing Statistic:\;Sy_t' = \alpha y_t + (1-\alpha)...
- 3
1
vote
1
answer
47
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Which topological spaces does every ultrafilter on the clopens converge?
Compact spaces are exactly those in which every ultrafilter on $\mathcal{P}X$ converges, or equivalently ultrafilters on the frame of opens $\mathcal{O}X$. In Stone spaces, or more generally compact $...
- 932
1
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1
answer
57
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Toy exercise on ultrafilters for $\mathbb{N}$
I'm reading something that defines ultrafilters like this.
An $\text{ultrafilter}$ on $\mathbb{N}$ is a subset $\omega$ of $2^{\mathbb{N}} - \{\varnothing\}$ so that
(Completeness) For any $A \subset ...
- 484
3
votes
1
answer
175
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Is Hahn-Banach equivalent to the ultrafilter lemma in ZF
I know that the ultrafilter lemma is weaker than the axiom of choice (in ZF)
And that in order to prove Choice in ZF from the ultrafilter lemma we need the Krein-Milman theorem
so $UF+KM=AC$
...
- 716
5
votes
2
answers
220
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Filter/Ultrafilter Methods in Commutative Algebra/ ring theory
I have a rather broad question: Are there any useful results/ approaches known on study commutative unitary rings and their ideals with filter/ ultrafilter methods?
What I know:
In Boolean algebra ...
