Prenormality of ideals and completeness of their quotient algebras

Morawiec, A.; Węglorz, B.

Colloquium Mathematicae, Tome 66 (1993), p. 19-27 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Publié le : 1993-01-01

Zbl 0814.04004

EUDML-ID : urn:eudml:doc:210167

@article{bwmeta1.element.bwnjournal-article-cmv64i1p19bwm,
     author = {A. Morawiec and B. W\k eglorz},
     title = {Prenormality of ideals and completeness of their quotient algebras},
     journal = {Colloquium Mathematicae},
     volume = {66},
     year = {1993},
     pages = {19-27},
     zbl = {0814.04004},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv64i1p19bwm}
}

Morawiec, A.; Węglorz, B. Prenormality of ideals and completeness of their quotient algebras. Colloquium Mathematicae, Tome 66 (1993) pp. 19-27. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv64i1p19bwm/

Bibliographie

[000] [BTW] J. E. Baumgartner, A. D. Taylor and S. Wagon, Structural properties of ideals, Dissertationes Math. 197 (1982). | Zbl 0549.03036

[001] [CWZ] J. Cichoń, B. Węglorz and R. Zrotowski, Some properties of filters. II, preprint no. 12, Mathematical Institute, Wrocław University, 1984.

[002] [J] T. Jech, Set Theory, Academic Press, New York 1978.

[003] [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.

[004] [U] S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150. | Zbl 56.0920.04

[005] [Z1] R. Zrotowski, A characterization of normal ideals, Abstracts Amer. Math. Soc. 4 (5) (1983), 386, no. 83T-03-381.

[006] [Z2] R. Zrotowski, personal communication.