Skip to main content
Springer Nature Link
Log in

The Categorical Equivalence Between Domains and Interpolative Generalized Closure Spaces

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Closure space has been proven to be a useful tool to restructure lattices and various order structures. This paper aims to provide an approach to characterizing domains by means of closure spaces. The notion of an interpolative generalized closure space is presented and shown to generate exactly domains, and the notion of an approximable mapping between interpolative generalized closure spaces is identified to represent Scott continuous functions between domains. These produce a category equivalent to that of domains with Scott continuous functions. Meanwhile, some important subclasses of domains are discussed, such as algebraic domains, L-domains, bounded-complete domains, and continuous lattices. Conditions are presented which, when fulfilled by an interpolative generalized closure space, make the generated domain fulfill some restrictive conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramsky, S., Domain theory in logical form, Annals of Pure and Applied Logic 51:1–77, 1991.

    Article  Google Scholar 

  2. Abramsky, S., and A. Jung, Domain theory, Oxford University Press, Oxford, 1994.

    Google Scholar 

  3. Birkhoff, G., Rings of sets, Duke Mathematical Journal 3(3):443–454, 1937.

    Article  Google Scholar 

  4. Birkhoff, G., Lattice theory, AMS Colloquium Publications, Providence, Rhode Island, 3nd edition, 1973.

  5. Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2nd edition, 2002.

  6. Erné, M., General Stone duality, Topology and its Applications 137:125–158, 2004.

    Article  Google Scholar 

  7. Erné, M., Closure. Closure, in F. Mynard, and E. Pearl, (eds.), Beyond Topology, vol 486 of Contemporary Mathematics, American Mathematical Society, Providence, 2009, pp. 163–238.

  8. Ganter, B., and R. Wille, Formal Concept Analasis, Springer-Verlag, 1999.

  9. Gierz, G., K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, 2003.

  10. Goubault-Larrecq, J., Non-Hausdorff Topology and Domain Theory, vol 22 of New Mathematical Monographs, Cambridge University Press, 2013.

  11. Guo, L., and Q. Li, The categorical equivalence between algebraic domains and F-augmented closure spaces, Order 32:101–116, 2015.

    Article  Google Scholar 

  12. Hitzler, P., M. Kröetzsch, and G. Zhang., A categorical view on algebraic lattices in formal concept analysis, Fundamenta Informaticae 74:1–29, 2006.

    Google Scholar 

  13. Hoofman, R., Continuous information systems, Information and Computation 105: 42–71, 1993.

    Article  Google Scholar 

  14. Huang, M., X. Zhou, and Q. Li, Re-visiting axioms of information systems, Information and Computation 247:130–140, 2016.

  15. Johnstone, P. T., Stone Spaces, Cambridge University Press, 1982.

  16. Jung, A., M. Kegelmann, and M. A. Moshier, Multi lingual sequent calculus and coherent spaces, Fundamenta Informaticae 37:369–412, 1999.

    Article  Google Scholar 

  17. Lawson, J. D., The round ideal completion via sobrification, Topology Proceedings 22:261–274, 1997.

    Google Scholar 

  18. Larsen, K. G., and G. Winskel, Using information systems to solve recursive domain equations effectively, in G. Kahn, D.B. MacQueen, and G. Plotkin, (eds.), Semantics of Data Types, vol 173 of Lecture Notes in Computer Science, Springer Verlag, 1984, pp. 109–130.

  19. Ledda, A., Stone-type representations and dualities for varieties of bilattices, Studia Logica 106(2):417–448, 2018.

    Article  Google Scholar 

  20. Li, Q., L. Wang, and L. Yao, A representation of continuous lattices based on closure spaces, Quaestiones Mathematicae 44(11):1513–1528, 2021.

    Article  Google Scholar 

  21. Mac Lane, S.,  Categories for the Working Mathematician, vol 5 of Graduate Texts in Mathematics, Springer Verlag, 1971.

  22. Mislove, M., Topology, domain theory and theoretical computer science, Topology and Its Applications 89: 3–59, 1998.

    Article  Google Scholar 

  23. Ranzato, F., Closures on CPOs form complete lattices, Information and Computation 152:236–249, 1999.

    Article  Google Scholar 

  24. Scott, D. S., Domains for Denotational Semantics, in M. Nielsen, and E.M. Schmidt, (eds.), International Colloquium on Automata, Languages and Programs, vol. 140 of Lecture Notes in Computer Science, Springer Verlag,1982, pp. 577–613.

  25. Spreen, D., L. Xu, and X. Mao, Information systems revisted—the general continuous case, Theoretical Computer Science 405:176–187, 2008.

  26. Spreen, D., Generalised information systems capture L-domains, Theoretical Computer Science 869:1–28, 2021.

    Article  Google Scholar 

  27. Stone, M.H., The theory of representations for Boolean algebras, Transactions of the American Mathematical Society 40:37–111, 1936.

    Google Scholar 

  28. Wang, L., and Q. Li, A representation of proper BC domains based on conjunctive sequent calculi, Mathematical Structures in Computer Science 30:1–13, 2020.

    Article  Google Scholar 

  29. Wang, L., and Q. Li, A logic for Lawson compact algebraic L-domains, Theoretical Computer Science 813:410–427, 2020.

    Article  Google Scholar 

  30. Wang, L., and Q. Li., Representations of stably continuous semi-lattices by information systems and abstract bases., Information Processing Letters 165:106036, 2021.

    Article  Google Scholar 

  31. Wang, L., Q. Li and X. Zhou, Continuous L-domains in logical form, Annals of Pure and Applied Logic 172:102993, 2021.

  32. Wu, M., L. Guo, and Q. Li, New representations of algebraic domains and algebraic L-domains via closure systems, Semigroup Forum 103:700–712, 2021.

    Article  Google Scholar 

  33. Xu, L., and X. Mao, When do abstract bases generate continuous lattices and \(L\)-domains, Algebra Universalis 58(1):95–104, 2008.

    Article  Google Scholar 

  34. Zhang, G., and G. Shen, Approximable concepts, Chu space, and information systems, Theory and Applications of Categories 17(5):80–102, 2006.

    Google Scholar 

  35. Zhao, D. S., Closure spaces and completions of posets, Semigroup Forum 90(2): 545–555, 2015.

    Article  Google Scholar 

Download references

Acknowledgements

We would like to give thanks to the anonymous referees for their comments and suggestions, which are of great help to improve this paper. This paper was supported by the National Natural Science Foundation of China (12231007) and Shandong Provincial Natural Science Foundation (ZR2022MA022).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China

    Longchun Wang

  2. School of Mathematics, Hunan University, Changsha, 410082, Hunan, China

    Qingguo Li

Authors
  1. Longchun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingguo Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qingguo Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Presented by Yde Venema

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, L., Li, Q. The Categorical Equivalence Between Domains and Interpolative Generalized Closure Spaces. Stud Logica 111, 187–215 (2023). https://doi.org/10.1007/s11225-022-10024-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-022-10024-3

Keywords