### Existence of endo-rigid Boolean Algebras

by Shelah. [Sh:229]

Around classification theory models, 1986

In [Sh:89] we, answering a question of Monk, have explicated the
notion of ``a Boolean algebra with no endomorphisms except the ones
induced by ultrafilters on it'' (see section 2 here) and proved the
existence of one with character density aleph_0, assuming first
diamondsuit_{aleph_1} and then only CH . The idea was that if
h is an endomorphism of B, not among the ``trivial'' ones, then
there are pairwise disjoint D_n in B with h(d_n) not subset
d_n . Then we can, for some S subset omega, add an element x
such that d <= x for n in S, x cap d_n=0 for n not in S
while forbidding a solution for {y cap h(d_n):n in S} cup {y cap
h(d_n)=0:n not in S} . Further analysis showed that the point is
that we are omitting positive quantifier free types. Continuing
this Monk succeeded to prove in ZFC, the existence of such Boolean
algebras of cardinality 2^{aleph_0} .
We prove (in ZFC) the existence of such B of density character
lambda and cardinality lambda^{aleph_0} whenever
lambda > aleph_0 . We can conclude answers to some questions from
Monk's list. We use a combinatorial method from [Sh:45],[Sh:172],
that is represented in Section 1.

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