Every ultrafilter falls into exactly one of two categories: principal or free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, each principal ultrafilter is of the form for some element of the given poset. In this case is called the of the ultrafilter. Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter. For arbitrary , the set is a filter, called the principal filter at ; it is a principal ultrafilter only if it is maximal.
Every ultrafilter falls into exactly one of two categories principal or free A principal (or fixed or trivial) ultrafilter is a filter containing a least element Consequently each principal ultrafilter is of the form for some element of the given poset In this case is called the of the ultrafilter Any ultrafilter that is not principal is called a free (or non-principal) ultrafilter For arbitrary  the set is a filter called the principal filter at ; it is a principal ultrafilter only if it is maximal