Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Every $(\lambda^+,\varkappa^+)$-regular ultrafilter is $(\lambda,\varkappa)$-regular

Author: Paolo Lipparini
Journal: Proc. Amer. Math. Soc. 128 (2000), 605-609
MSC (1991): Primary 03C20, 04A20
Published electronically: July 8, 1999
MathSciNet review: 1623032
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following:

Theorem A. If $D$ is a $(\lambda^+,\varkappa)$-regular ultrafilter, then either

$D$ is $(\lambda,\varkappa)$-regular, or
the cofinality of the linear order $\prod _D\langle\lambda,<\rangle$ is $\operatorname{cf}\varkappa$, and $D$ is $(\lambda,\varkappa')$-regular for all $\varkappa'<\varkappa$.

Corollary B. Suppose that $\varkappa$ is singular, $\varkappa>\lambda$ and either $\lambda$ is regular, or $\operatorname{cf}\varkappa<\operatorname{cf}\lambda$. Then every $(\lambda^{+n},\varkappa)$-regular ultrafilter is $(\lambda,\varkappa)$-regular.

We also discuss some consequences and variations.

References [Enhancements On Off] (What's this?)

  • [BK] M. Benda and J. Ketonen, Regularity of ultrafilters, Israel J. Math. 17 (1974), 231-240. MR 53:132
  • [CC] G. V. Cudnovskii and D. V. Cudnovskii, Regular and descending incomplete ultrafilters (English Translation), Soviet Math. Dokl. 12 (1971), 901-905.
  • [CK] C. C. Chang and H. J. Keisler, Model Theory, North Holland, Amsterdam, 1977. MR 58:27177
  • [Ka] A. Kanamori, Weakly normal filters and irregular ultrafilters, Trans Amer. Math. Soc. 220 (1976), 393-399. MR 58:240
  • [KM] A. Kanamori and M. Magidor, The evolution of large cardinal axioms in Set Theory, in Higher Set Theory, Lecture Notes in Mathematics, 669, Springer-Verlag, Berlin, 1978. MR 80b:03083
  • [Kei] J. Keisler, On cardinalities of ultraproducts, Bull. Amer. Math. Soc. 70 (1964), 644-647.
  • [Ket] J. Ketonen, Nonregular ultrafilters and large cardinals, Trans. Amer. Math. Soc. 224 (1976), 61-73. MR 54:7260
  • [KP] K. Kunen and K. L. Prikry, On descendingly incomplete ultrafilters, J. Symbolic Logic 36 (1971), 650-652. MR 46:1585
  • [Lp1] P. Lipparini, More on regular ultrafilters in ZFC, revised for J. Symbol. Logic.
  • [Lp2] P. Lipparini, Ultrafilter translations, I: $(\lambda ,\lambda )$-compactness of logics with a cardinality quantifier, Arch. Math. Logic 35 (1996), 63-87. MR 96k:03087

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 03C20, 04A20

Retrieve articles in all journals with MSC (1991): 03C20, 04A20

Additional Information

Paolo Lipparini
Affiliation: Dipartimento di Matematica, Viale della Ricerca Scientifica, II Università di Roma (Tor Vergata), I-00133 Rome, Italy

Keywords: $(\alpha, \varkappa)$-regular ultrafilter, cofinality of ultrapowers, almost $(\lambda, \varkappa)$-regular extensions
Received by editor(s): November 20, 1997
Received by editor(s) in revised form: April 8, 1998
Published electronically: July 8, 1999
Additional Notes: This work was performed under the auspices of G.N.S.A.G.A
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society