Sh:229

• Shelah, S. (1986). Existence of endo-rigid Boolean algebras. In Around classification theory of models, Vol. 1182, Springer, Berlin, pp. 91–119. arXiv: math/9201238 DOI: 10.1007/BFb0098506 MR: 850054
  Part of [Sh:d]
• Abstract:
  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 §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 $d\leq x$ 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.

• Version 1996-03-11_10 (19p) published version (29p)

@incollection{Sh:229,
 author = {Shelah, Saharon},
 title = {{Existence of endo-rigid Boolean algebras}},
 booktitle = {{Around classification theory of models}},
 series = {Lecture Notes in Math.},
 volume = {1182},
 year = {1986},
 pages = {91--119},
 publisher = {Springer, Berlin},
 mrnumber = {850054},
 mrclass = {06E05 (03C65)},
 doi = {10.1007/BFb0098506},
 note = {\href{https://arxiv.org/abs/math/9201238}{arXiv: math/9201238} Part of [Sh:d]},
 arxiv_number = {math/9201238},
 refers_to_entry = {Part of [Sh:d]}
}