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A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness


Author: Matatyahu Rubin
Journal: Trans. Amer. Math. Soc. 278 (1983), 65-89
MSC: Primary 06E05; Secondary 03E35, 03G05
MathSciNet review: 697061
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Abstract: Using $ {\diamondsuit_{{\aleph_1}}}$ we construct a Boolean algebra $ B$ of power $ {\aleph_1}$, with the following properties: (a) $ B$ has just $ {\aleph_1}$ subalgebras. (b) Every uncountable subset of $ B$ contains a countable independent set, a chain of order type $ \eta $, and three distinct elements $ a,b$ and $ c$, such that $ a \cap b = c$. (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. $ B$ is embeddable in $ P(\omega )$. A variant of the construction yields an almost Jónson Boolean algebra. We prove that every subalgebra of an interval algebra is retractive. This answers affirmatively a conjecture of B. Rotman. Assuming MA or the existence of a Suslin tree we find a retractive BA not embeddable in an interval algebra. This refutes a conjecture of B. Rotman. We prove that an uncountable subalgebra of an interval algebra contains an uncountable chain or an uncountable antichain. Assuming CH we prove that the theory of Boolean algebras in Magidor's and Malitz's language is undecidable. This answers a question of M. Weese.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0697061-6
Article copyright: © Copyright 1983 American Mathematical Society