Abstract
In Bezhanishvili et al. (2012) we introduced the category MKHaus of modal compact Hausdorff spaces, and showed these were concrete realizations of coalgebras for the Vietoris functor on compact Hausdorff spaces, much as modal spaces are coalgebras for the Vietoris functor on Stone spaces. Also in Bezhanishvili et al. (2012) we introduced the categories MKRFrm and MDV of modal compact regular frames, and modal de Vries algebras as algebraic counterparts to modal compact Hausdorff spaces, much as modal algebras are algebraic counterparts to modal spaces. In Bezhanishvili et al. (2012), MKRFrm and MDV were shown to be dually equivalent to MKHaus, hence equivalent to one another. Here we provide a direct, choice-free proof of the equivalence of MKRFrm and MDV. We also detail connections between modal compact regular frames and the Vietoris construction for frames (Johnstone 1982, 1985), discuss a Vietoris construction for de Vries algebras, and how it is linked to modal de Vries algebras. Also described is an alternative approach to the duality of MKRFrm and MKHaus obtained by using modal de Vries algebras as an intermediary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramsky, S.: A Cook’s tour of the finitary non-well-founded sets. In: Artemov, S. et al. (ed.) We Will Show Them: Essays in honour of Dov Gabbay. College Publications, pp. 1–18 (2005)
Banaschewski, B.: The duality of distributive continuous lattices. Canad. J. Math. 32(2), 385–394 (1980)
Banaschewski, B.: Compactification of frames. Math. Nachr. 149, 105–115 (1990)
Banaschewski, B., Mulvey, C.J.: Stone-Čech compactification of locales. I. Houston J. Math. 6(3), 301–312 (1980)
Bezhanishvili, G.: Stone duality and Gleason covers through de Vries duality. Topology Appl. 157(6), 1064–1080 (2010)
Bezhanishvili, G.: De Vries algebras and compact regular frames. Appl. Categ. Struct. 20, 569–582 (2012)
Bezhanishvili, G., Bezhanishvili, N., Harding, J.: Modal compact Hausdorff spaces. J. Log. Comput. (2012). doi:10.1093/logcom/exs030
Bezhanishvili, G., Harding, J.: Proximity frames and regularization. Appl. Categ. Struct. (2013). doi:10.1007/s10485-013-9308-9
Bezhanishvili, N., Kurz, A.: Free modal algebras: a coalgebraic perspective. In: Algebra and Coalgebra in Computer Science. Lecture Notes in Comput. Sci., vol. 4624, pp. 143–157. Springer, Berlin (2007)
Celani, S., Jansana, R.: Priestley duality, a Sahlqvist theorem and a Goldblatt-Thomason theorem for positive modal logic. Log. J. IGPL 7(6), 683–715 (1999)
de Vries, H.: Compact spaces and compactifications. An algebraic approach. Ph.D. thesis, University of Amsterdam (1962)
Dowker, C.H.: Mappings of proximity structures. In: General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961), pp. 139–141. Academic Press, New York (1962)
Frith, J.L.: The category of uniform frames. Cahiers Topologie Géom. Différentielle Catég. 31(4), 305–313 (1990)
Ghilardi, S.: An algebraic theory of normal forms. Ann. Pure Appl. Logic 71(3), 189–245 (1995)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)
Halmos, P.R.: Algebraic Logic. Chelsea Publishing Co., New York (1962)
Hofmann, K.H., Lawson, J.D.: The spectral theory of distributive continuous lattices. Trans. Amer. Math. Soc. 246, 285–310 (1978)
Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)
Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)
Johnstone, P.T.: Vietoris locales and localic semilattices. In: Continuous lattices and their applications (Bremen, 1982). Lecture Notes in Pure and Appl. Math., vol. 101, pp. 155–180. Dekker, New York (1985)
Naimpally, S.A., Warrack, B.D.: Proximity Spaces. Cambridge Tracts in Mathematics and Mathematical Physics, no. 59. Cambridge University Press, London (1970)
Picado, J., Pultr, A.: Frames and Locales. Frontiers in Mathematics, Birkhäuser/Springer Basel AG, Basel (2012)
Smirnov, Y.M.: On proximity spaces. Mat. Sbornik N.S. 31(73), 543–574, (1952) (Russian)
Smyth, M.B.: Stable compactification. I. J. London Math. Soc. 45(2), 321–340 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bezhanishvili, G., Bezhanishvili, N. & Harding, J. Modal Operators on Compact Regular Frames and de Vries Algebras. Appl Categor Struct 23, 365–379 (2015). https://doi.org/10.1007/s10485-013-9332-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-013-9332-9