## Prenormality of ideals and completeness of their quotient algebras

### Volume 64 / 1993

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.