8
THOMAS JECH AND KAREL PRIKRY
If I is prime then sat(I) = 2. If sat(I) is a finite number then K
is the union of finitely many atoms of I (a set X £ I is an atom if X/I
is an atom of the Boolean algebra B),and I is essentially a combination of
finitely many prime ideals.
If sat(I) is infinite, then by [5], sat(I) is a regular uncountable
cardinal. Clearly, every ideal I over K is (2 ) -saturated, and so if I
is atomless then sat(I) is a regular cardinal satisfying
(1.5) ^ sat(I) 5
(2K)+.
If K is real-valued measurable, then the ideal (1.2) is ^1-saturated.
If K carries a K-saturated ideal then K is weakly inaccessible (in fact
weakly Mahlo), see [15]. If K carries a K -saturated ideal I, then
carries a normal K -saturated ideal J such that sat(J) 5 sat(I), cf. [15].
Existence of a K -saturated ideal over K does not necessarily entail that
K is a limit cardinal: Kunen constructed a model in which tf carries an
^-saturated ideal.
The ideal of thin sets is not K-saturated: by a theorem of Solovay in
[15], every stationary subset of K is the union of K pairwise disjoint
stationary subsets. In section 4 we address ourselves to the question whether
the ideal of thin sets can be K -saturated (see the remark following Theorem
3.2.1.)
1.3. Two sets X £ K and Y £ K are almost disjoint if |X D Y| K. Fam-
ilies of almost disjoint sets have been investigated by Sierpinski, Tarski,
and more recently by Baumgartner [1]. It is clear that the question of size
of almost disjoint families of sets X £ K of size K is equivalent to the
problem of evaluating sat(I) for the ideal I = {X £ K : |X| K}.
It is easy to construct an almost disjoint family of K subsets of K
+ a
size K and so sat(I) K . If 2 K for all a K, then there exists
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