Duals of subdirectly irreducible modal algebras
Yde Venema

Abstract:
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.