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

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.