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

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

## 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.

Item Type: Report PP-2002-10 Prepublication (PP) Series 2002 boolean algebras with operators, modal logic, subdirectly irreducible algebras, simple algebras, duality Logic 12 Oct 2016 14:36 12 Oct 2016 14:36 https://eprints.illc.uva.nl/id/eprint/74