Ultrafilter

An ultrafilter is a set theoretic structure.

Definition

An ultrafilter on a set $X$ is a non-empty filter $\mathcal{F}$ on $X$ with the following property:

For every set $A \subseteq X$ , either $A$ or its complement is an element of $\mathcal{F}$ .

An ultrafilter is a finest filter, that is, if $\mathcal{F}$ is an ultrafilter on $X$ , then there is no filter $\mathcal{F'}$ on $X$ such that $\mathcal{F} \subsetneq \mathcal{F'}$ . All finest filters are also ultrafilters; we will prove this later.

Types of Ultrafilters

An ultrafilter is said to be principle, or fixed, or trivial if it has a least element, i.e., an element which is a subset of all the others. Otherwise, an ultrafilter is said to be nontrivial, or free, or non-principle.

Proposition. Let $\mathcal{F}$ be a trivial ultrafilter on $X$ . Then there exists an element $a\in X$ such that $\mathcal{F}$ is the set of subsets of $X$ which contain $a$ .

Proof. Let $A$ be a minimal element of $\mathcal{F}$ . It suffices to show that $A$ contains a single element. Indeed, let $a$ be an element of $A$ . Since $\mathcal{F}$ is an ultrafilter, one of the sets $\{ a\}$ , $X \setminus \{a\}$ must be an element of $\mathcal{F}$ . But $A \not\subseteq X \setminus \{a\}$ , so $\{a\}$ must be an element of $\mathcal{F}$ . Hence $A \subseteq \{a\}$ , so $A = \{a\}$ , as desired. $\blacksquare$

Evidently, the only filters on finite sets are trivial.

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Ultrafilter

Definition

Types of Ultrafilters

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