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We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters; among them:  (a) If  m  ≥ 1 and the ultrafilter  D  is (  m  ( λ  + n  ),   m  ( λ  + n  ))‐regular, then  D  is  κ  ‐decomposable for some  κ  with  λ  ≤  κ  ≤ 2  λ   (Theorem 4.3(a  '  )).  (b) If  λ  is a strong limit cardinal and  D  is (  m  ( λ  + n  ),   m  ( λ  + n  ))‐regular, then either  D  is (cf  λ , cf  λ )‐regular or there are arbitrarily large  κ  <  λ  for which  D  is  κ  ‐decomposable (Theorem 4.3(b)).  (c) Suppose that  λ  is singular,  λ  <  κ , cf  κ  ≠ cf  λ  and  D  is ( λ  + ,  κ )‐regular. Then:  (i)  D  is either (cf  λ , cf  λ )‐regular, or ( λ '  ,  κ )‐regular for some  λ '   <  λ  (Theorem 2.2).  (ii) If  κ  is regular, then  D  is either ( λ ,  κ )‐regular, or ( ω ,  κ  ' )‐regular for every  κ '   <  κ  (Corollary 6.4).  (iii) If either (1)  λ  is a strong limit cardinal and  λ <λ   < 2  κ  , or (2)  λ <λ   <  κ , then  D  is either  λ  ‐decomposable, or ( λ '  ,  κ )‐regular for some  λ '   <  λ  (Theorem 6.5).  (d) If  λ  is singular,  D  is ( μ , cf  λ )‐regular and there are arbitrarily large  ν  <  λ  for which  D  is  ν  ‐decomposable, then  D  is  κ  ‐decomposable for some  κ  with  λ  ≤  κ  ≤  λ  < μ   (Theorem 5.1; actually, our result is stronger and involves a covering number).  (e)  D  ×  D  '  is ( λ ,  μ )‐regular if and only if there is a  ν  such that  D  is ( ν ,  μ )‐regular and  D '   is ( λ ,  ν  ' )‐regular for all  ν ∼   <  ν  (Proposition 7.1).  We also list some problems, and furnish applications to topological spaces and to extended logics (Corollar‐ies 4.6 and 4.8) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

More on regular and decomposable ultrafilters in ZFC