Definition: In mathematics, an ultrafilter is a type of filter that captures all sets in its domain. An ultrafilter on a set A is defined as a collection of subsets of A that are closed under finite intersection and complementation (i.e., for any two non-empty intersections of elements from the collection, their complements are also in the collection). An ultrafilter on a set A can be thought of as an ideal that captures all sets that are "too small" to be included in A.

ultrafilterable

Definition: Ultrafilterability, also known as ultrafilter completeness, is a mathematical property that refers to the ability of an infinite set of non-empty sets (or sub-sets) to be well-ordered by inclusion. This means that there exists a well-ordering $<$ such that every two elements in the set are comparable under this order. In other words, if we have a finite set $A$ and an infinite set $B$, then it is possible to find a maximal element of $B

ultrafilterability

Definition: In mathematics, an ultrafilter on a set X is a subset of X that satisfies the following properties: 1. The union of all elements in the filter equals X. 2. If x is any element not in the filter, then x must be in the filter. For example, consider the set {x ∈ R | x < 3} and the ultrafilter generated by this set (the set of all numbers less than 3). The union of all elements in this filter equals R

ultrafilter

Definition: A non-ultrafilterable is a concept in computer science and information theory. A non-ultrafilterable set is a property that cannot be expressed as an ultrafilter, but can be expressed as a family of ultrafilters on the same set. In other words, it's like saying "I have a list of shoes, I want to filter out only the ones that fit my size, but I don't want to use any filters at all." This way, you're not limited

nonultrafilterable