Abstract
The concepts of hypercontinuous posets and generalized completely continuous posets are introduced. It is proved that for a poset P the following three conditions are equivalent: (1) P is hypercontinuous; (2) the dual of P is generalized completely continuous; (3) the normal completion of P is a hypercontinuous lattice. In addition, the relational representation and the intrinsic characterization of hypercontinuous posets are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adámek, J., Herrllich, H., Strecker, G. E., Abstract and Concrete Categories, Wiley Interscience, New York, 1990.
Crawley, P. and Dilworth, R. P., Algebraic Theory of Lattices, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973.
Dedekind, R., Stetigkeit und Irrationale Zahlen, Vieweg, Braunschweig, 1872.
Erné, M., A Completion-Invariant Extension of the Concept of Continuous Lattices, Banaschewski B. and Hoffmann R. -E. (eds.), Continuous Lattices, Proc. Bremen, 1979, Lecture Notes in Math., Vol. 871, Springer-Verlag, Berhn, Heidelberg, New York, 1981, 43–60.
Erné, M., The Dedekind-MacNeille completion as a reflector, Order, 8, 1991, 159–173.
Erné, M., Bigeneration in complete lattices and principal separation in ordered sets, Order, 8, 1991, 197–221.
Erné, M., Distributive laws for concept lattices, Algebra Universalis, 30, 1993, 538–580.
Frink, O., Ideals in partially ordered sets, Amer. Math. Monthly, 61, 1954, 223–234.
Gierz, G., Hofmann, K. H., Keimel, K, et al., Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003.
Gierz, G. and Lawson, J. D., Generalized continuous lattices and hypercontinuous lattices, Rocky Mountain J., 11, 1981, 271–296.
Harding, J. and Bezhanishvili, G., Macneille completions of heyting algebras, Houston Journal of Mathematics, 30, 2004, 937–952.
MacNeille, H. M., Partially ordered sets, Trans. Amer. Math. Soc., 42, 1937, 416–460.
Niederle, J., On infinitely distributive ordered sets, Math. Slovaca., 55, 2005, 495–502.
Raney, G., A subdirect union representation for completely distributive complete lattices, Proc. Amer. Math. Soc., 4, 1953, 518–522.
Venugopolan, P., A generalization of completely disdributive lattices, Algebra Universalis, 27, 1990, 578–586.
Xu, X. Q. and Liu, Y. M., Relational Representations of Hypercontinuous Lattices, Domain Theory, Logic and Computation, Kluwer Academic Publishers, 2003, 65–74.
Yang, J. B. and Xu, X. Q., The dual of a generalized completely distributive lattice is a hypercontinuous lattice, Algebra Universalis, 63, 2010, 145–149.
Zhang, W. F. and Xu, X. Q., Completely precontinuous posets, Electronic Notes in Theoretical Computer Science, 301, 2014, 169–178.
Zhang, W. F. and Xu, X. Q., Meet precontinuous posets, Electronic Notes in Theoretical Computer Science, 301, 2014, 179–188.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Nos. 10861007, 11161023), the National Excellent Doctoral Dissertation of China (No. 2007B14), the Ganpo 555 Programme for Leading Talents of Jiangxi Province, the Natural Science Foundation of Jiangxi Province (No. 20114BAB201008) and the Fund of Education Department of Jiangxi Province (No. GJJ12657).
Rights and permissions
About this article
Cite this article
Zhang, W., Xu, X. Hypercontinuous posets. Chin. Ann. Math. Ser. B 36, 195–200 (2015). https://doi.org/10.1007/s11401-015-0913-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-015-0913-9