Abstract
This is a short survey illustrating some of the essential aspects of the theory of canonical extensions. In addition some topological results about canonical extensions of lattices with additional operations in finitely generated varieties are given. In particular, they are doubly algebraic lattices and their interval topologies agree with their double Scott topologies and make them Priestley topological algebras.
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References
Alo, R.A., Frink, O.: Topologies of lattice products. Canad. J. Math. 18, 1004–1014 (1966)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)
Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001)
Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Amer. Math. Soc. 358, 573–590 (2005)
Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Japonica 40, 207–215 (1994)
Gehrke, M., Jónsson, B.: Monotone bounded distributive lattice expansions. Math. Japonica 52, 197–213 (2000)
Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94, 13–45 (2004)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)
Harding, J.: On profinite completions and canonical extensions. Alg. Universalis 55, 293–296 (2006)
Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Amer. J. Math. 73, 891–939 (1951)
Vosmaer, J.: Logic, Algebra and Topology. Investigations into canonical extensions, duality theory and point-free topology, Ph.D Dissertation, University of Amsterdam (2010)
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Gehrke, M., Vosmaer, J. (2011). A View of Canonical Extension. In: Bezhanishvili, N., Löbner, S., Schwabe, K., Spada, L. (eds) Logic, Language, and Computation. TbiLLC 2009. Lecture Notes in Computer Science(), vol 6618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22303-7_6
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DOI: https://doi.org/10.1007/978-3-642-22303-7_6
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