Finest partitions for ultrafilters

Publisher： Cambridge University Press

E-ISSN： 1943-5886|51|2|327-332

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.51, Iss.2, 1986-06, pp. : 327-332

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Abstract

If a uniform ultrafilter U over an uncountable cardinal κ is not outright countably complete, probably the next best thing is that it have a finest partition: a master function f:κ → ω with ƒ −({n}) ∉ U for each n ϵ ω such that for any g: κ → κ, either (a) it is one-to-one on a set in U, or (b) it factors through ƒ (mod U), i.e. for some function h, {α < κ ∣ h(f(α)) = g(α)} ϵ U. In this paper, it is shown that recent contructions of irregular ultrafilters over ω 1 can be amplified to incorporate a finest partition.

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