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, or, topological relational structure. These characterizations involve a special binary topo-reachability relation on the dual structure; we call a point u a topo-root of the dual structure if every ultrafilter is topo-reachable from u. We prove that a boolean algebra with operators is simple iff every point in the dual structure is a topo-root; and that it is subdirectly irreducible iff the collection of topo-roots is open and non-empty in the Stone topology on the dual structure iff this collection has non-empty interior in that topology.
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References
Esakia, L.n ‘To the theory of modal and superintuitionistic systems’ in V. Smirnov, (ed.), Proceedings of the USSR Symposium on Logical Inference, Nauka, Moscow, 1979, pp. 147-172. (In Russian).
Goldblatt, R. I., ‘Varieties of complex algebras’, Annals of Pure and Applied Logic 38:173-241, 1989.
Kracht, M., Tools and Techniques in Modal Logic, Number 142 in Studies in Logic, Elsevier, Amsterdam, 1999.
Sambin, G., ‘Subdirectly irreducible modal algebras and initial frames’, Studia Logica 62:269-282, 1999.
Sambin, G., and V. Vaccaro, ‘Topology and duality in modal logic’, Annals of Pure and Applied Logic 37:249-296, 1988.
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Venema, Y. A Dual Characterization of Subdirectly Irreducible BAOs. Studia Logica 77, 105–115 (2004). https://doi.org/10.1023/B:STUD.0000034188.80692.46
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DOI: https://doi.org/10.1023/B:STUD.0000034188.80692.46