Ultrafilter - Ordering On Ultrafilters

Ordering On Ultrafilters

Rudin–Keisler ordering is a preorder on the class of ultrafilters defined as follows: if U is an ultrafilter on X, and V an ultrafilter on Y, then if and only if there exists a function f: X → Y such that

for every subset C of Y.

Ultrafilters U and V are Rudin–Keisler equivalent, if there exist sets, and a bijection f: A → B which satisfies the condition above. (If X and Y have the same cardinality, the definition can be simplified by fixing A = X, B = Y.)

It is known that is the kernel of, i.e., if and only if and .