闭刚性布尔代数的存在性
Existence of endo-rigid Boolean algebras
摘要：
在[Sh：89]中，我们回答了蒙克的一个问题，阐明了“除了由超滤子引起的自同态之外没有自同态的布尔代数”的概念（见第2节），并证明了一个具有特征密度aleph_0的布尔代数的存在，首先假设diamondsuit_{aleph_1}，然后只假设CH。其思想是，如果h是B的自同态，而不是“平凡的”自同态，那么B中有成对不相交的D_n，h（d_n）不是子集d_n。然后，我们可以对某个S子集Ω，添加一个元素x，使得对于S中的n，d <= x，对于不在S中的n，x cap d_n=0，同时禁止{y cap h（d_n）：n in S} cup {y cap h（d_n）=0：n not in S}的解。进一步的分析表明，关键是我们忽略了正量词自由类型。继续这蒙克成功地证明了ZFC，存在这样的布尔代数基数2^{aleph_0}。我们证明（在ZFC）存在这样的B的密度字符λ和基数λ ^{aleph_0}时，λ> aleph_0。我们可以从蒙克的清单中总结出一些问题的答案。我们使用[Sh：45]，[Sh：172]中的组合方法，该方法在第1节中表示。
英文摘要：
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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