# Prenormality of ideals and completeness of their quotient algebras

Colloquium Mathematicae (1993)

- Volume: 64, Issue: 1, page 19-27
- ISSN: 0010-1354

## Access Full Article

top## Abstract

top## How to cite

topMorawiec, A., and Węglorz, B.. "Prenormality of ideals and completeness of their quotient algebras." Colloquium Mathematicae 64.1 (1993): 19-27. <http://eudml.org/doc/210167>.

@article{Morawiec1993,

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.},

author = {Morawiec, A., Węglorz, B.},

journal = {Colloquium Mathematicae},

keywords = {ideals on cardinals; prenormal ideal; quotient Boolean algebra},

language = {eng},

number = {1},

pages = {19-27},

title = {Prenormality of ideals and completeness of their quotient algebras},

url = {http://eudml.org/doc/210167},

volume = {64},

year = {1993},

}

TY - JOUR

AU - Morawiec, A.

AU - Węglorz, B.

TI - Prenormality of ideals and completeness of their quotient algebras

JO - Colloquium Mathematicae

PY - 1993

VL - 64

IS - 1

SP - 19

EP - 27

AB - 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.

LA - eng

KW - ideals on cardinals; prenormal ideal; quotient Boolean algebra

UR - http://eudml.org/doc/210167

ER -

## References

top- [BTW] J. E. Baumgartner, A. D. Taylor and S. Wagon, Structural properties of ideals, Dissertationes Math. 197 (1982). Zbl0549.03036
- [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. Zbl56.0920.04
- [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.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.