Prenormality of ideals and completeness of their quotient algebras

A. Morawiec; B. Węglorz

Colloquium Mathematicae (1993)

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

Abstract

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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.

How to cite

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Morawiec, 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

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  1. [BTW] J. E. Baumgartner, A. D. Taylor and S. Wagon, Structural properties of ideals, Dissertationes Math. 197 (1982). Zbl0549.03036
  2. [CWZ] J. Cichoń, B. Węglorz and R. Zrotowski, Some properties of filters. II, preprint no. 12, Mathematical Institute, Wrocław University, 1984. 
  3. [J] T. Jech, Set Theory, Academic Press, New York 1978. 
  4. [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. 
  5. [U] S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre, Fund. Math. 16 (1930), 140-150. Zbl56.0920.04
  6. [Z1] R. Zrotowski, A characterization of normal ideals, Abstracts Amer. Math. Soc. 4 (5) (1983), 386, no. 83T-03-381. 
  7. [Z2] R. Zrotowski, personal communication. 

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