Prenormality of ideals and completeness of their quotient algebras

Volume 64 / 1993

A. Morawiec, B. Węglorz Colloquium Mathematicum 64 (1993), 19-27 DOI: 10.4064/cm-64-1-19-27

Abstract

It is well known that if a nontrivial ideal ℑ on κ is normal, its quotient Boolean algebra P(κ)/ℑ is $κ^+$-complete. It is also known that such completeness of the quotient does not characterize normality, since P(κ)/ℑ turns out to be $κ^+$-complete whenever ℑ is prenormal, i.e. whenever there exists a minimal ℑ-measurable function in $^{κ}κ$. Recently, it has been established by Zrotowski (see [Z1], [CWZ] and [Z2]) that for non-Mahlo κ, not only is the above condition sufficient but also necessary for P(κ)/ℑ to be $κ^+$-complete. In the present note we are going to visualize that Zrotowski's result is a consequence of the Boolean structure of P(κ) exclusively, rather than of its other particular properties.

Authors

  • A. Morawiec
  • B. Węglorz

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