# 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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### When the restriction of a maximal filter is maximal? [closed]

Let P be a distributive lattice and Q one of its sublattices. Let F be a maximal filter in P, thus F interesecated with Q is a filter in Q. But, when it is maximal?
1 vote
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### Not closed under ultraproduct, then not closed under "shorter" ultraproduct

The problem Let $L$ be a first-order logic, $K$ a class of $L$-structures closed under elementary equivalence and $\kappa$ the cardinality of the set containing all formulas of $L$. Prove that if $K$ ...
1 vote
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### Class of countably join-complete lattices is not closed under ultraproduct

I need to prove the above claim by constructing a counterexample. My intuition is that $\mathcal{A}_n = \langle A, \leq \rangle$ where $A = (0, 1, \ldots, n)$ should work, but cannot verify why. Any ...
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### 1D power spectrum of field filtered by 3D filters

Consider a 3D scalar function $u(x,y,z)$, in a numerical implementation represented by discrete values $u_{i,j,k}$ on a Cartesian grid. If this scalar field is filtered by a 1D filter, either ...
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### Taking the ultrapower of the real numbers as an ordered field twice.

I've seen ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$ discussed in this question, with the $\mathbb{R}$ case being described as the more straightforward case. I'm wondering what happens when you ...
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### Ultraproduct with respect to a partially ordered set

I am interested in learning about limits with respect to ultrafilters on a poset (partially ordered set) and ultraproducts with respect to a poset. However, all I can find about this topic only ...
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### Calculating the Jacobian with respect to pixel intensity for Gaussian Blur

I'm applying a Gaussian blur to an image where each pixel of the original image is being optimized for some purpose and the Gaussian blur is an intermediate transformation on the pixel intensities. It ...
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### Prove that the forgetful functor $U: \textbf{Ab} \to \textbf{Set}$ preserves all filtered colimits

I realize that my question is exactly the same as this post here. However, I tried finding the book that was mentioned, Borceux's Handbook of Category Theory I, but my efforts to find the book here ...
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### How can I prove the existence of delta-incomplete / countable incomplete ultrafilters?

I am reading a W.A.J Luxemburg 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-...
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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 ...
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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 ...
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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 ...
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