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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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