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

Abstract

Using ♢ ℵ 1 {\diamondsuit _{{\aleph _1}}} we construct a Boolean algebra B B of power ℵ 1 {\aleph _1} , with the following properties: (a) B B has just ℵ 1 {\aleph _1} subalgebras. (b) Every uncountable subset of B B contains a countable independent set, a chain of order type η \eta , and three distinct elements a , b a,b and c c , such that a ∩ b = c a \cap b = c . (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. B B is embeddable in P ( ω ) 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 \text {B} . Rotman. Assuming MA \text {MA} or the existence of a Suslin tree we find a retractive BA \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 CH \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.