On completeness of the quotient algebras ${\cal P}(\kappa)/I$

Abstract.

In this paper, the following are proved:

Theorem A. The quotient algebra ${\cal P} (\kappa )/I$ is complete if and only if the only non-trivial I -closed ideals extending I are of the form $I\lceil A$ for some $A\in I^+$ .

Theorem B. If $\kappa$ is a stationary cardinal, then the quotient algebra ${\cal P} (\kappa )/ NS_\kappa$ is not complete.

Corollary. (1) If $\kappa$ is a weak compact cardinal, then the quotient algebra ${\cal P} (\kappa )/NS_\kappa$ is not complete.

(2) If $\kappa$ bears $\kappa$ -saturated ideal, then the quotient algebra ${\cal P} (\kappa )/NS_\kappa$ is not complete.

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 ${\cal P} (\kappa )/I$ is complete, it is B -valid that for some $A\subseteq\check\kappa$ , ${\cal P} (\kappa )/({\bf J}\lceil A)$ is complete, where J is the ideal generated by $\check I$ in $V^B$ .

Corollary. 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.