On completeness of the quotient algebras {\cal P}(\kappa)/I |
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Authors: | Yasuo Kanai |
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Institution: | (1) Toyota National College of Technology, 2-1 Eisei-cho Toyota, Aichi 471-8525, Japan (e-mail:Kanai@toyota-ct.ac.jp) , JP |
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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.
Received: 1 April 1997 / Revised version: 1 July 1998 |
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Keywords: | |
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