Ultrafilter mappings and their Dedekind cuts

Blass, Andreas

doi:10.1090/S0002-9947-1974-0351822-6

Ultrafilter mappings and their Dedekind cuts
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Trans. Amer. Math. Soc. 188 (1974), 327-340 Request permission

Abstract:

Let D be an ultrafilter on the set N of natural numbers. To each function $p:N \to N$ and each ultrafilter E that is mapped to D by p, we associate a Dedekind cut in the ultrapower D-prod N. We characterize, in terms of rather simple closure conditions, the cuts obtainable in this manner when various restrictions are imposed on E and p. These results imply existence theorems, some known and some new, for various special kinds of ultrafilters and maps.

References

Orderings of ultrafilters

Andreas Blass, The Rudin-Keisler ordering of $P$-points, Trans. Amer. Math. Soc. 179 (1973), 145–166. MR 354350, DOI 10.1090/S0002-9947-1973-0354350-6
David Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970/71), no. 1, 1–24. MR 277371, DOI 10.1016/0003-4843(70)90005-7
Gustave Choquet, Construction d’ultrafiltres sur N, Bull. Sci. Math. (2) 92 (1968), 41–48 (French). MR 234405
Gustave Choquet, Deux classes remarquables d’ultrafiltres sur N, Bull. Sci. Math. (2) 92 (1968), 143–153 (French). MR 236860
P. Erdös, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294. MR 19911, DOI 10.1090/S0002-9904-1947-08785-1
P. Erdös and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470. MR 1556929
T. Frayne, A. C. Morel, and D. S. Scott, Reduced direct products, Fund. Math. 51 (1962/63), 195–228. MR 142459, DOI 10.4064/fm-51-3-195-228
H. Jerome Keisler, Ultraproducts and saturated models, Nederl. Akad. Wetensch. Proc. Ser. A 67 = Indag. Math. 26 (1964), 178–186. MR 0168483, DOI 10.1016/S1385-7258(64)50021-9
Mary Ellen Rudin, Partial orders on the types in $\beta N$, Trans. Amer. Math. Soc. 155 (1971), 353–362. MR 273581, DOI 10.1090/S0002-9947-1971-0273581-5
Walter Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956), 409–419. MR 80902