On completeness of the quotient algebras \({\mathcal P}(\kappa)/I\)
type: Item
instance of: scholarly article
title: On completeness of the quotient algebras \({\mathcal P}(\kappa)/I\)
zbMATH Open document ID: 0954.03055
author: Yasuo Kanai
published in: Archive for Mathematical Logic
publication date: 2000-12-14T00:00:00Z
review text: The author calls an ideal \(I\) on \(\kappa\) completive if its quotient algebra \(\wp (\kappa)/I\) is complete. Then, he gives a characterization of completive ideals and applies it to the following question: Can one always get completeness of the quotient algebra \(\wp(\kappa)/NS_\kappa\)? His main results are as follows: Theorem A. \(I\) is completive if and only if the only non-trivial \(I\)-closed ideals extending \(I\) are of the form \(I\lceil A\) for some \(A\in I^+\), where an ideal \(J\) is said to be \(I\)-closed if: whenever \(A\) is a subset of \(J\) and \([X]_I=\bigvee_{Y\in A}[Y]_I\), \(X\) is in \(J\). Theorem B. The non-stationary ideal on any stationary cardinal is not completive. On the other hand, he also gets some consistency results, that is, using a general preservation lemma he presents a model of ZFC which separates completeness and \(\kappa^+\)-saturation of the quotient algebra \(\wp(\kappa)/I\). Theorem C. Assume that \(\kappa\) is a strongly compact cardinal, \(I\) is a non-trivial normal \(\kappa\)-complete ideal on \(\kappa\) and \(B\) is an \(I\)-regular complete Boolean algebra. Then if \(I\) is completive, it is \(B\)-valid that for some \(A\subseteq\check\kappa\), \(J\lceil A\) is completive, where \(J\) is the ideal generated by \(\check I\) in \(V^B\). Corollary 1. Let \(M\) be a transitive model of ZFC and in \(M\) let \(\kappa\) be a strongly compact cardinal and \(\lambda\) a regular uncountable cardinal less than \(\kappa\). Then there exists a generic extension \(M[G]\) in which \(\kappa=\lambda^+\) and \(\kappa\) carries a non-trivial \(\kappa\)-complete ideal \(I\) which is completive but not \(\kappa^+\)-saturated. Corollary 2. If ZFC + ``there is a strongly compact cardinal'' is consistent, then so is ZFC + ``there is a regular uncountable cardinal \(\kappa\) which bears a non-trivial \(\kappa\)-complete ideal \(I\) such that the quotient algebra \(\wp(\kappa)/I\) is complete but not \(\kappa^+\)-saturated.'' Although, t…
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 06E10
zbMATH DE Number: 1443414
zbMATH Keywords: completeness of \(\wp(\kappa)/I\)
zbMATH Keywords: ideal
zbMATH Keywords: stationary cardinal
zbMATH Keywords: consistency
MaRDI profile type: Publication
full work available at URL: s001530050004
OpenAlex ID: W2054745748
DOI: 10.1007/S001530050004
description: scientific article; zbMATH DE number 1443414