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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness
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by Matatyahu Rubin PDF
Trans. Amer. Math. Soc. 278 (1983), 65-89 Request permission

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.
References
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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