(Q1571831)

Statements

instance of

mathematical concept

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subclass of

filter

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image

Filter vs ultrafilter.svg
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media legend

in the powerset lattice of {1,2,3,4}, the principal filter ↑{1,4} (dark green) isn’t an ultrafilter since it can be extended to the larger nontrivial filter ↑{1} by including also the light green elements; but ↑{1} is an ultrafilter since it can’t be so extended (English)

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studied in

lattice theory

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defining formula

${\displaystyle \varnothing \neq \bigcup _{x\in U}\uparrow x=U\subsetneq P\land (\forall x,y\in U\exists z\in U\colon x\geq z\leq y)\land \forall F\subsetneq P\colon F\supset U\land \left(\varnothing \neq \bigcup _{x\in F}\uparrow x=F\subset P\land (\forall x,y\in F\exists z\in F\colon x\geq z\leq y)\right)\implies F=U}$

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in defining formula

${\displaystyle (P,\leq )}$

symbol represents

partially ordered set

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${\displaystyle U}$

symbol represents

ultrafilter

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${\displaystyle \varnothing }$

symbol represents

empty set

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maintained by WikiProject

WikiProject Mathematics

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Stack Exchange tag

https://mathoverflow.net/tags/ultrafilters

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different from

Q12377736

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Identifiers