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## Colloquium Mathematicum

1993 | 64 | 1 | 19-27
Tytuł artykułu

### Prenormality of ideals and completeness of their quotient algebras

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Treść / Zawartość
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EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
19-27
Opis fizyczny
Daty
wydano
1993
otrzymano
1988-11-30
poprawiono
1990-05-25
Twórcy
autor
• Mathematical Institute, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
autor
• Mathematical Institute, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
• [BTW] J. E. Baumgartner, A. D. Taylor and S. Wagon, Structural properties of ideals, Dissertationes Math. 197 (1982).
• [CWZ] J. Cichoń, B. Węglorz and R. Zrotowski, Some properties of filters. II, preprint no. 12, Mathematical Institute, Wrocław University, 1984.
• [J] T. Jech, Set Theory, Academic Press, New York 1978.
• [S] R. Solovay, Real-valued measurable cardinals, in: Axiomatic Set Theory (Proc. Univ. of California, Los Angeles, Calif., 1967), Proc. Sympos. Pure Math. 13, Part I, Amer. Math. Soc., Providence, R.I., 1971, 397-428.
• [U] S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150.
• [Z1] R. Zrotowski, A characterization of normal ideals, Abstracts Amer. Math. Soc. 4 (5) (1983), 386, no. 83T-03-381.
• [Z2] R. Zrotowski, personal communication.
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