Existence of endo-rigid Boolean algebras

Saharon Shelah

Published 1992-01-15Version 1

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.