A dual characterization of subdirectly irreducible BAOs (Q1876093)

Given a Boolean algebra with operators \({\mathbf{A}} = ({\mathbf{B}}_{{\mathbf{A}}} , (f_i)_{i \in I})\), a relation \(R^{\star}\), called the {topo-reachability relation}, is defined on the set \(A_*\) of ultrafilters of \({\mathbf{B}}_{{\mathbf{A}}}\). If \(T_{{{\mathbf{A}}}_*}\) is the collection of those ultrafilters \(u\) of \({\mathbf{B}}_{{\mathbf{A}}}\) such that \(R^{\star}[u] = A_*\), it is proved that \({\mathbf{A}}\) is simple iff \(T_{{\mathbf{A}}_*} = A_*\), and \({\mathbf{A}}\) is subdirectly irreducible iff \(T_{{\mathbf{A}}_*}\) is non-empty and open in \(A_*\) with the Stone topology.