Definition:Principal Ultrafilter

Definition
Let $X$ be a set, and $\mathcal P \left({X}\right)$ be the power set of $X$.

Let $\mathcal F \subset \mathcal P \left({X}\right)$ be an ultrafilter on $X$ with a cluster point.

Then $\mathcal F$ is a principal ultrafilter or a fixed ultrafilter on $X$.

Nonprincipal Ultrafilter
If $\mathcal F \subset \mathcal P \left({X}\right)$ is an ultrafilter on $X$ which does not have a cluster point, it is a nonprincipal ultrafilter or a free ultrafilter on $X$.

Also see

 * From Ultrafilter has One Cluster Point at Most, an ultrafilter is either principal or nonprincipal - there are no other options.


 * In Principal Ultrafilter is All Sets Containing Cluster Point it is shown that a principal ultrafilter whose cluster point is $x$ is the set of all subsets of $X$ which contain $x$.