Perfect extensions of regular double Stone algebras (Q1902544)

If \(A\) and \(B\) are regular double Stone algebras, then \(B\) is a perfect extension of \(A\) if (i) \(A\) is a subalgebra of \(B\) and \(B\) is complete and atomic, (ii) whenever \(\{x_i/i \in I\}\) is a family of elements of \(A\) such that \(\sum x_i = 1\) then \(\sum \{x_j/j \in J\} = 1\) for a finite subset \(J\) of \(I\), and (iii) whenever \(u,v\) are join-irreducible elements of \(B\) with \(v \not \leq u\), then there exists \(a \in A\) with \(a \geq u\) and \(v \not \leq a\). The author shows the existence and uniqueness of perfect extensions of regular double Stone algebras. The result is extended to regular double Stone algebras with operators. He also shows that every regular double Stone algebra is isomorphic to an algebra of rough subsets of an approximation space.