Abstract
The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone’s duality for distributive lattices rather than Priestley’s. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper.
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References
Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)
Birkhoff G.: Rings of sets. Duke Math. J. 3, 443–454 (1937)
Gehrke M., Harding J.: Bounded lattice expansions. J. 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)
Halmos, P.: Algebraic Logic, I. Monadic Boolean algebras. Compos. Math. 12, 217–249 (1956)
Hartonas, C.: Duality for lattice-ordered algebras and for normal algebraizable logics. Studia Logica 58, 403-450 (1997)
Hartung, G.: A topological representation for lattices. Algebra Universalis 17, 273–299 (1992)
Hofmann, K. H., Mislove, M. W., and Stralka, A. R.: The Pontryagin Duality of Compact 0-Dimensional Semilattices and Its Applications. Lecture Notes in Math., vol. 396. Springer (1974)
Jónsson, B. and Tarski, A.: Boolean algebras with operators. I. Amer. J. Math. 73, 891–939 (1951)
Moshier, M. A. and Jipsen, P.: Topological Duality and Lattice Expansions Part II: Lattice Expansions with Quasioperators. Algebra Universalis (in press).
Nachbin, L.: On a characterization of the lattice of all ideals of a Boolean ring. Fund. Math. 36, 137–142 (1949)
Priestley H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)
Stone M.H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111 (1936)
Stone, M. H.: Topological representation of distributive lattices. Casopsis pro Pestovani Matematiky a Fysiky 67, 1–25 (1937)
Tarski, A.: Sur les classes closes par rapport à certaines opérations élémentaires. Fund. Math. 16, 195–197 (1929)
Urquhart, A.: A topological representation theorem for lattices. Algebra Universalis 8, 45–58 (1978)
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Moshier, M.A., Jipsen, P. Topological duality and lattice expansions, I: A topological construction of canonical extensions. Algebra Univers. 71, 109–126 (2014). https://doi.org/10.1007/s00012-014-0267-2
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DOI: https://doi.org/10.1007/s00012-014-0267-2