A Characterization Theorem for Continuous Lattices by Closure Spaces

  • Guozhi Ma
  • Lankun GuoEmail author
  • Cheng Yang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11511)


The notion of closure spaces plays an important role in formal concept analysis, and there exists a close connection between formal concept analysis and lattice theory. In order to restructure continuous lattices, a special kind of complete lattices in Domain theory, this paper proposes a novel notion named relationally consistent F-augmented closure spaces. Then, the concept of F-approximable mappings between relationally consistent F-augmented closure spaces is introduced, which provides a representation of Scott continuous maps between continuous lattices. The final result is: the categories of relationally consistent F-augmented closure spaces and continuous lattices are equivalent.


Closure space Approximable mapping Continuous lattice 


  1. 1.
    Barr, M., Wells, C.: Category Theory for Computing Science, 3rd edn. Prentice Hall, New York (1990)zbMATHGoogle Scholar
  2. 2.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order, 2nd edn. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  3. 3.
    Erné, M.: Lattice representations for categories of closure spaces, Categorical topology (Toledo, OH, 1983). Bentley, H.L. (ed.) Sigma Series in Pure Mathematics, vol. 5, pp. 197–222. Heldermann, Berlin (1984)Google Scholar
  4. 4.
    Erné, M.: General stone duality. Topology Appl. 137(1–3), 125–158 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ers̆ov, Y.L.: The theory of A-spaces. Algebra Logic 12(4), 209–232 (1973)CrossRefGoogle Scholar
  6. 6.
    Escardó, M.H.: Properly injective spaces and function spaces. Topology Appl. 89(1), 75–120 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ganter, B., Wille, R.: Formal Concept Analysis: Mathematical Foundations. Springer, Berlin (1999). Scholar
  8. 8.
    Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  9. 9.
    Guo, L., Li, Q.: The categorical equivalence between algebraic domains and F-augmented closure spaces. Order 32, 101–116 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hofmann, K.H., Keimel, K.: A General Character Theory for Partially Osets and Lattices, vol. 122, no. 122. Memoirs of the American Mathematical Society (1972)Google Scholar
  11. 11.
    Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Scott, D.: Continuous lattices. In: Lawvere, F.W. (ed.) Toposes, Algebraic Geometry and Logic. LNM, vol. 274, pp. 97–136. Springer, Heidelberg (1972). Scholar
  13. 13.
    Stone, M.H.: The theory of representations of Boolean algebras. Trans. Am. Math. Soc. 40, 37–111 (1936)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Engineering and DesignHunan Normal University, ChangshaHunanPeople’s Republic of China
  2. 2.College of Mathematics and Statistics, Key Laboratory of High Performance, Computing and Stochastic Information ProcessingHunan Normal University, ChangshaHunanPeople’s Republic of China

Personalised recommendations