Abstract
The main result of this paper is a generalization of the classical equivalence between the category of continuous posets and the category of completely distributive lattices, based on the fact that the continuous posets are precisely the spectra of completely distributive lattices. Here we show that for so-called hereditary and union complete subset selections Z, the category of Z-continuous posets is equivalent (via a suitable spectrum functor) to the category of Z-supercompactly generated lattices; these are completely distributive lattices with a join-dense subset of certain Z-hypercompact elements. By appropriate change of the morphisms, these equivalences turn into dualities. We present two different approaches: the first one directly uses the Z-join ideal completion and the Z-below relation; the other combines two known equivalence theorems, namely a topological representation of Z-continuous posets and a general lattice theoretical representation of closure spaces.
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Banaschewski, B. and Hoffmann, R.-E. (eds.): Continuous Lattices, Bremen 1979, Lecture Notes in Math. 871, Springer-Verlag, Berlin, Heidelberg, New York, 1981.
Bandelt, H.-J. and Erné, M.: The category of Z-continuous posets, J. Pure Appl. Algebra 30 (1983), 219-226.
Bandelt, H.-J. and Erné, M.: Representations and embeddings of M-distributive lattices, Houston J. Math. 10 (1984), 315-324.
Erné, M.: Scott convergence and Scott topologies on partially ordered sets II, in [1], pp. 61-96.
Erné, M.: Homomorphisms of M-distributive and M-generated posets, Tech. Report 125, Institut für Mathematik, Universität Hannover, 1981.
Erné, M.: Adjunctions and standard constructions for partially ordered sets, in G. Eigenthaler et al. (eds.), Contributions to General Algebra 2, Proc. Klagenfurt Conf. 1982, Hölder-Pichler-Tempsky, Wien, 1983, pp. 77-106.
Erné, M.: Lattice representation of closure spaces, in L. Bentley et al. (eds.), Categorical Topology, Conf. Taledo, Ohio 1983, Heldermann Verlag, Berlin, 1984, pp. 197-222.
Erné, M.: Order extensions as adjoint functors, Quaestiones Math. 9 (1986), 146-204.
Erné, M.: Z-continuous posets, Z-ary closure spaces and generalized soberness, Seminar of Continuous Semilattices, Memo 10-5-1985 (Preprint).
Erné, M.: Z-continuity, Z-hypercompactness and complete distributivity, Seminar of Continuous Semilattices, Memo 6-4-1986 (Preprint).
Erné, M.: Algebraic ordered sets and their generalizations, in I. Rosenberg and G. Sabidussi (eds.), Algebras and Orders, Proc. Montreal 1992, Kluwer, 1994.
Erné, M.: The ABC of order and topology, in H. Herrlich and H.-E. Porst (eds.), Category Theory at Work, Heldermann Verlag, Berlin, 1991, pp. 57-83.
Erné, M.: Z-continuous posets and their topological manifestation, in Workshop Domains II, Informatik-Berichte 96-04, TU Braunschweig, 1996, pp. 53-88, and Appl. Cat. Structures 7 (1999), 31-70.
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S.: A Compendium of Continuous Lattices, Springer-Verlag, Berlin, Heidelberg, New York, 1980.
Hoffmann, R.-E.: Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications, in [1], pp. 159-208.
Hoffmann, R.-E. and Hofmann, K. H. (eds.): Continuous Lattices and Their Applications, Bremen 1982, Lecture Notes in Pure and Appl. Math. 101, Marcel Dekker, New York, 1985.
Hofmann, K. H. and Mislove, M.: Free objects in the category of completely distributive lattices, in [16], pp. 129-151.
Lawson, J. D.: The duality of continuous posets, Houston J. Math. 5 (1979), 357-386.
Markowsky, G.: A motivation and generalization of Scott's notion of a continuous lattice, in [1], pp. 298-307.
Meseguer, J.: Order completion monads, Algebra Universalis 16 (1983), 63-82.
Novak, D.: Generalization of continuous posets, Trans. Amer. Math. Soc. 272 (1982), 645-667.
Raney, G. N.: A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518-522.
Scott, D. S.: Continuous Lattices, in Algebraic Geometry and Logic, Lecture Notes in Math. 274, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
Wright, J. B., Wagner, E. G. and Thatcher, J. W.: A uniform approach to inductive posets and inductive closure, Theor. Comp. Sci. 7 (1978), 57-77.
Venugopalan, G.: Z-continuous posets, Houston J. Math. 12 (1986), 275-294.
Zhao, D.: N-compactness in L-fuzzy topological spaces, J. Math. Analysis and Appl. 128 (1987), 64-79.
Zhao, D.: Bases of completely distributive lattices, J. Fuzzy Math. 1 (1997), 103-109.
Zhao, D.: On projective Z-frames, Canad. Math. Bull. 40 (1997), 39-46.
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Erné, M., Zhao, D. Z-Join Spectra of Z-Supercompactly Generated Lattices. Applied Categorical Structures 9, 41–63 (2001). https://doi.org/10.1023/A:1008758815245
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DOI: https://doi.org/10.1023/A:1008758815245