In general for a class of subsets on natural numbers, if all properties of a free ultrafilter are met (inclusion of supersets, maximal, empty intersection of all sets, inclusion of cofinite sets) except that “closed under intersection” is replaced by “closed under intersection of A in the class and any cofinite set” must this class necessarily be a free ultrafilter? If not, are there examples of such incomplete ultrafilters that are not free ultrafilters in the standard sense.

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