Skip to main content

Essential and density topologies on s 2-continuous posets

  • CHONGXIA LU (a1) and QINGGUO LI (a1)

Recently, Rusu and Ciobanu established that for a continuous domain L, a subset B of L is a basis if and only if B is dense with respect to the d-topology, called the density topology, on L. In situations where directed completeness fails, Erné has proposed in 1991 an alternative definition of continuity called s 2-continuity which remedied the lack of stability of continuity under the classical Dedekind–MacNeille completion. In this paper, we show how the ‘Rusu–Ciobanu’ type of characterization can be formulated and established over the class of s 2-continuous posets with appropriate modifications. Although we obtain more properties of essential topologies and density topologies on s 2-continuous posets, respectively.

Hide All
Abramsky, S. and Jung, A. (1994). Domain theory. In: Abramsky, S., Gabbay, D.M., and Maibaum, T. S. E. (eds.) Semantic Structures, Handbook of Logic in Computer Science, vol. 3, Clarendon Press, Oxford, 1168.
Banaschewski, B. (1977). Essential extensions of T 0-spaces. General Topology and its Applications 7 (3) 233246.
Davey, B.A. and Priestley, H.A. (2002). Introduction to Lattices and Order, 2nd ed., Cambridge University Press, Cambridge.
Engelking, R. (1977). General Topology, Polish Scientific Publishers, Warszawa.
Erné, M. (1991). The ABC of order and topology. In: Herlich, H. and Porst, H.-E. (eds.) Category Theory at Work, Heldermann Verlag, Berlin, 5783.
Erné, M. (1991). The Dedekind-MacNeille completion as a reflector. Order 8 (2) 159173.
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M. and Scott, D.S. (2003). Continuous Lattices and Domains, Encyclopedia of Mathematics and Its Applications, vol. 93, Cambridge University Press, Cambridge.
Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory, New Mathematical Monographs, vol. 22, Cambridge University Press, Cambridge.
Hoffmann, R.-E. (1981). Continuous posets, prime spectra of completely distrbutive lattices and Hausdorff compactifications. In: Banaschewski, B. and Hoffmann, R.-E. (eds.) Continuous Lattices, Lecture Notes in Mathematics, vol. 871, Springer-Verlag, Berlin, 159208.
Rusu, D. and Ciobanu, G. (2016). Essential and density topologies of continuous domains. Annals of Pure and Applied Logic 167 (9) 726736.
Zhang, W.F. and Xu, X.Q. (2015). s 2-quasicontinuous posets. Theoretical Computer Science 574 7885.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 14 *
Loading metrics...

Abstract views

Total abstract views: 145 *
Loading metrics...

* Views captured on Cambridge Core between 30th October 2017 - 20th April 2018. This data will be updated every 24 hours.