I don”t understand the definition of principal ultrafilter

definitionfilters

I am very new to filters. Wikipedia says that principal ultrafilters are filters containing least element.

However, then it says principal ultrafilters look like $F_a = \{x: a \leq x \}$ for some elements $a$ of the given poset.

How can the $F_a$ contain the least element, if it consists of $x$ bigger than the $a$?

Also, how can principal ultrafilter have the condition that the intersection of all its subsets is the filter itself?

I think I generally have trouble understanding the definition of principal ultrafilters, since I just cannot see how this makes sense.

Thank you for any help.

Best Answer

This question seems to be about posets in general, but since your other question was about filters on sets specifically, I will preamble this answer that you could regard $(\mathcal P(X),\subseteq)$ as a partial order with minimal element $\varnothing$. Simply replace $P$ with $\mathcal P(X)$, $\leq$ with $\subseteq$ and $0$ with $\varnothing$ below if you're only interested in sets.

Let's fix some arbitrary poset $(P,\leq)$ and consider filters on $P$.

A filter $F$ is principal if it contains some least element, not the least element. What this means to say, is that there is some element in $F$ that is smaller than all other elements of $F$. For instance, $F_a=\{x\in P\mid a\leq x\}$ is principal, but it is not necessarily the case that $a$ is the least element of $P$, and if in fact $a$ happens to be the least element of $P$, then $F_a$ would not be an interesting filter: it would contain every element of $P$.

The filter $F_a$ contains all elements $x$ such that $x\geq a$, thus $a$ is the least element of $F_a$: first, $a\in F_a$ since $a\leq a$, and second, if $x\in F_a$, then $a\leq x$.

If $F$ is a principal filter, then it is not necessarily an ultrafilter. For example, if $P$ has a minimal element $0$, and furthermore $0<a<b$ are two elements of the poset, then $F_a\supset F_b$, but $a\notin F_b$. Both $F_a$ and $F_b$ are principal filters, but since $F_b$ is not maximal, it is not an ultrafilter.

An element $a\in P$ is called atomic if there is a minimal element $0\in P$ for which $0<a$ and for every $x\in P$, if $x<a$, then $x=0$. A principal filter $F$ is an ultrafilter if and only if $F=F_a$ for some atomic $a$:

Suppose $F=F_b$ for some non-atomic $b$, then there is some $0<x<b$, and thus $F_x\supset F_b$, which shows that $F_b$ is not maximal.

Conversely if $a$ is atomic and $F\supset F_a$ is a filter containing some $x\notin F_a$, then by filters being downward directed, there is some $y\leq x$ and $y\leq a$. Now $y\neq a$, since if $y=a$, then $a\leq x$ contradicts that $x\notin F_a$. But $y\neq a$ implies that $y=0$, since $a$ is atomic, hence $F_y=F_0=P$ is not a proper filter.

Not every ultrafilter has to be principal, but it is hard to give concrete examples of nonprincipal ultrafilters. It generally requires some form of the Axiom of Choice to prove their existence in the first place.

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