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A dual characterization of subdirectly irreducible BAOs. (English) Zbl 1050.03045
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.

MSC:
03G25 Other algebras related to logic
06E25 Boolean algebras with additional operations (diagonalizable algebras, etc.)
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