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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1983-0697061-6
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 $\text {B}$. Rotman. Assuming $\text {MA}$ or the existence of a Suslin tree we find a retractive $\text {BA}$ 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 $\text {CH}$ 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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Article copyright: © Copyright 1983 American Mathematical Society