Geodesic
Survey on the Tukey Theory of Ultrafilters
Zbornik radova, Tome 17 (2015) no. 25, p. 53 .

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This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin--Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.
Classification : 05-02 03-02 03E05 05C55 54D80
Keywords: ultrafilter, Tukey, Rudin-Keisler, cofinal type, Ramsey-classification 
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     author = {Natasha Dobrinen},
     title = {Survey on the {Tukey} {Theory} of {Ultrafilters}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZR_2015_17_25_a2/}
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Natasha Dobrinen. Survey on the Tukey Theory of Ultrafilters. Zbornik radova, Tome 17 (2015) no. 25, p. 53 . http://geodesic.mathdoc.fr/item/ZR_2015_17_25_a2/