On some algebra of sets in Steprāns strong-Q-sequence model

Let H ( A ) stand for the ideal of sets that hereditarily belong to a given algebra A of sets. Let S , S 0 denote the operations of Marczewski. We say that 〈 A , H ( A ) 〉 is inner MB-representable (is topological) if 〈 A , H ( A ) 〉 = 〈 S ( F ) , S 0 ( F ) 〉 for a nonempty F ⊂ A (for F equal to τ ∖ { ∅ } where τ is a topology on ⋃ A ). We show that the existence of an algebra A isomorphic to P ( ω ) , with 〈 A , H ( A ) 〉 inner MB-representable but not topological, and with complete Boolean algebra A / H ( A ) , is independent of ZFC.