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

Using (FORMULA PRESENTED) we construct a Boolean algebra B of power ᵳ₁ with the following properties: (a) B has just ᵳ₁ subalgebras, (b) Every uncountable subset of B contains a countable independent set, a chain of order type n, and three distinct elements a, b and c, such that a∩b = c. (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. B is embeddable in P(ω). A variant of the construction yields an almost Jônson 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.