On completeness of the quotient algebras \({\mathcal P}(\kappa)/I\)
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, the assumption of his consistency results is much stronger than that of \textit{A. Kanamori} and \textit{S. Shelah}'s results in Trans. Am. Math. Soc. 347, No. 6, 1963-1979 (1995; Zbl 0827.06009), his lemma gives a somewhat general method to preserve completeness. And the proof of Theorem B also presents a general method to obtain non-completive ideals. That is: Let \(\langle S, \prec \rangle\) be a partially ordered set and let \(\psi(v_0)\) be a formula of set theory with one free variable such that \(I_S=\{X \subseteq S:\psi(X)\}\) and \(I_a=\{X \subseteq\text{pr}(a):\psi (X)\}\) are proper ideals on \(S\) and \(\text{pr}(a)\) for each \(a\in \widetilde S\) respectively, where \(\text{pr} (a)=\{b\in S:b\prec a\}\), \(\widetilde S=\{a \in S:\text{cf(pr}(a))> \aleph_0\}\) and \(\text{cf(pr}(a))\) means a suitable cofinality of \(\text{pr}(a)\). Assume that: i) if \(t\) is in \((I_a)^*\), then \(\{b\in t\cap \widetilde S:t\cap \text{pr}(b)\) is in \((I_b)^+\}\) is also in \((I_a)^*\), ii) if \(X\in I^+_S\) and for each \(a\in X\) \(t_a\) is in \(I^+_a\), then \(\bigcup_{a\in X}\) \(t_a\) is in \(I^+_S\), iii) there is a subset \(A\) of \(S\) such that for any \(X\in I^+_S\), \(M_\psi(X)\cap A\) is in \(I^+_S\), where \(M_\psi(X)= \{a\in\widetilde S:X\cap \text{pr}(a)\) is in \(I^+_a \}\), and iv) for any \(X\in I^+_S\), \(X-M_\psi(X)\) is in \(I^+_S\). Then \(\wp(S)/I_S\) is not complete.
scientific article; zbMATH DE number 1443414