Abstract
The main objective of this paper (the second of two parts) is to show that quasioperators can be dealt with smoothly in the topological duality established in Part I. A quasioperator is an operation on a lattice that either is join preserving and meet reversing in each argument or is meet preserving and join reversing in each argument. The paper discusses several common examples, including orthocomplementation on the closed subspaces of a fixed Hilbert space (sending meets to joins), modal operators ◊ and □ on a bounded modal lattice (preserving joins, resp. meets), residuation on a bounded residuated lattice (sending joins to meets in the first argument and meets to meets in the second). This paper introduces a refinement of the topological duality of Part I that makes explicit the topological distinction between the duals of meet homomorphisms and of join homomorphisms. As a result, quasioperators can be represented by certain continuous maps on the topological duals.
Similar content being viewed by others
References
Dunn J.M., Gehrke M., Palmigiano A.: Canonical extensions and relational completeness of some substructural logics. J. Symbolic Logic 70, 713–740 (2005)
Gehrke M.: Generalized Kripke frames. Studia Logica, 84(2), 241–275 (2006)
Gehrke M., Harding J.: Bounded lattice expansions. J. of Algebra 238, 345–371 (2001)
Gierz G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A Compendium of Continuous Lattices. Springer (1980)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93. Cambridge Univ. Press, Cambridge (2003)
Hartonas, C.: Duality for lattice-ordered algebras and for normal algebraizable logics. Studia Logica 58, 403–450 (1997)
Hartonas C., Dunn J.M.: Stone duality for lattices. Algebra Universalis 37, 391–401 (1997)
Moshier, M.A., Jipsen, P.: Topological duality and lattice expansions Part I: A topological construction of canonical extensions. Algebra Universalis (in press).
Stone M.H.: Topological representation of distributive lattices. Casopsis pro Pestovani Matematiky a Fysiky 67, 1–25 (1937)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by M. Haviar.
Rights and permissions
About this article
Cite this article
Moshier, M.A., Jipsen, P. Topological duality and lattice expansions, II: Lattice expansions with quasioperators. Algebra Univers. 71, 221–234 (2014). https://doi.org/10.1007/s00012-014-0275-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-014-0275-2