# PP-2002-10: Duals of subdirectly irreducible modal algebras

PP-2002-10: Venema, Yde (2002) Duals of subdirectly irreducible modal algebras. [Report]

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We give a characterization of the simple, and of the subdirectly irreducible boolean algebras with operators (including modal algebras), in terms of the dual descriptive frame. These characterizations involve a special binary \emph{quasi-reachability} relation on the dual structure; we call a point $u$ a quasi-root of the dual structure if every ultrafilter is quasi-reachable from $u$. We prove that a boolean algebra with operators is simple iff every point in the dual structure is a quasi-root; and that it is subdirectly irreducible iff the collection of quasi-roots has measure nonzero in the Stone topology on the dual structure.