More homogeneous almost disjoint families. (Q1771918)

S. Shelah and J. Steprans showed the consistency of the existence of an uncountable almost disjoint family \(\mathcal A\) on \(\omega \) which generates a Boolean algebra \(A\) (i.e.\ if \(B\) is an uncountable subalgebra of \(A\) and \(\forall n \in \omega \) it holds \(\{ n \} \in B\) then \(A\) is isomorphic to \(B\)). The author proves a generalization of the previous result. The following is consistent: there exists an uncountable almost disjoint family \(\mathcal A\) on \(\omega \) which generates a Boolean algebra isomorphic to every uncountable (weak) subalgebra.