Partition subalgebras for maximal almost disjoint families (Q1849859)

Let \({\mathcal M}\) be a maximal almost disjoint family on \(\omega\). A partitioner of \({\mathcal M}\) is a set \(A\subseteq\omega\) such that the sets \({\mathcal X}=\{M\in {\mathcal M}:M \subseteq^*A\}\) and \({\mathcal Y}=\{M\in {\mathcal M}:M\cap A=^* \emptyset\}\) form a partition of \({\mathcal M}\). The set of all partitioners of \({\mathcal M}\) forms a Boolean algebra \({\mathcal A}\). Let \(I_{\mathcal M}\) denote the ideal in \({\mathcal A}\) which is generated by \({\mathcal M}\) together with the finite subsets of \(\omega\). The partitioner algebra \(P_{\mathcal M}\) of \({\mathcal M}\) is the quotient algebra \({\mathcal A}/I_{\mathcal M}\). A Boolean algebra is representable if it is isomorphic to a partitioner algebra of some mad family on \(\omega\). Previously, \textit{J. E. Baumgartner} and \textit{M. Weese} [``Partition algebras for almost disjoint families'', Trans. Am. Math. Soc. 274, 619-630 (1982; Zbl 0515.03032)] asked if every subalgebra of a representable algebra is also representable. It is known that the answer is yes under CH. Here the authors show that the existence of a supercompact cardinal implies that it is consistent that the answer is no. The proof uses \(\text{PFA}^+\) and \(\diamondsuit (S^2_0)\).