## Journal of Symbolic Logic

### Normality and $\mathscr{P}(\kappa)/\mathscr{J}$

R. Zrotowski

#### Abstract

The main result of this paper is that if $\kappa$ is not a weakly Mahlo cardinal, then the following two conditions are equivalent: 1. $\mathscr{P}(\kappa)/ \mathscr{J}$ is $\kappa^+$-complete. 2. $\mathscr{J}$ is a prenormal ideal. Our result is a generalization of an announcement made in [Z]. We say that $\mathscr{J}$ is selective iff for every $\mathscr{J}$-function $f: \kappa \rightarrow \kappa$ there is a set $X \in \mathscr{J}$ such that $f\mid(\kappa - X)$ is one-to-one. Our theorem provides a positive partial answer to a question of B. Weglorz from [BTW, p. 90], viz.: is every selective ideal $\mathscr{J}$, with $\mathscr{P}(\kappa)/\mathscr{J} \kappa^+$-complete, isomorphic to a normal ideal? The theorem is also true for fine ideals on $\lbrack\lambda\rbrack^{<\kappa}$ for any $\kappa \leq \lambda$, i.e. if $\kappa$ is not a weakly Mahlo cardinal then the Boolean algebra $\mathscr{P} (\lbrack\lambda\rbrack^{<\kappa})/\mathscr{J}$ is $\lambda^+$-complete iff $\mathscr{J}$ is a prenormal ideal (in the sense of $\lbrack\lambda\rbrack^{<\kappa})$.

#### Article information

Source
J. Symbolic Logic, Volume 56, Issue 3 (1991), 1064-1067.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743751

Digital Object Identifier
doi:10.2178/jsl/1183743751

Mathematical Reviews number (MathSciNet)
MR1129167

Zentralblatt MATH identifier
0743.03034

JSTOR
Zrotowski, R. Normality and $\mathscr{P}(\kappa)/\mathscr{J}$. J. Symbolic Logic 56 (1991), no. 3, 1064--1067. doi:10.2178/jsl/1183743751. https://projecteuclid.org/euclid.jsl/1183743751