Distributivity of the algebra of regular open subsets of \(\beta\mathbb R\setminus \mathbb R\) (Q1779224)

The algebra \(RO(\beta\mathbb R\setminus\mathbb R)\) of regular open subsets of the Čech-Stone remainder of the real line is isomorphic to the completion of the Boolean algebra \(A^\omega/\text{Fin}\) where \(A\)~is the Boolean algebra of clopen subsets of the Cantor set~\(2^\omega\) and \(\text{Fin}\) is the set of elements of~\(A^\omega\) with finite support. The authors prove that the distributivity number of the algebra \(A^\omega/\text{Fin}\) is below the distributivity number~\(\mathfrak h\) of the algebra \(\mathcal P(\omega)/\text{fin}\) and below the additivity of the ideal of meager sets of reals and they prove that the tower number of \(A^\omega/\text{Fin}\) is equal to the tower number~\(\mathfrak t\) of \(\mathcal P(\omega)/\text{fin}\). As a~corollary they obtain simple arguments for a~result of A.~Dow who proved that in the iterated Mathias model the spaces \(\beta\omega\setminus\omega\) and \(\beta\mathbb R\setminus\mathbb R\) are not co-absolute. The authors also prove that under the assumption \(\mathfrak t=\mathfrak h\) the spaces \(\beta\omega\setminus\omega\) and \(\beta\mathbb R\setminus\mathbb R\) are co-absolute which improves the same result of E.~van Douwen under the assumption \(\mathfrak p=\mathfrak h\).