SpringerLink
Log in

Representations for the Largest Extension of a Closure System

  • Published:
Order Aims and scope Submit manuscript

Abstract

An extension of a closure system on a finite set S is a closure system on the same set S containing the first one as a sublattice. A closure system can be represented in different ways, e.g. by an implicational base or by the set of its meet-irreducible elements. When a closure system is described by an implicational base, we provide a characterization of the implicational base for the largest extension. We also show that the largest extension can be handled by a small modification of the implicational base of the input closure system. This answers a question asked in Yoshikawa et al. (J. Math. Psychol. 77, 82–93, 2017). Second, we are interested in computing the largest extension when the closure system is given by the set of all its meet-irreducible elements. We give an incremental polynomial time algorithm to compute the largest extension of a closure system, and left open whether the number of meet-irreducible elements grows exponentially.

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
EUR 32.99 /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

Data Availibility

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Adaricheva, K.V., Nation, J.B.: Largest extension of a finite convex geometry. Algebra Universalis 52(2), 185–195 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Adaricheva, K.V., Gorbunov, V.A., Tumanov, V.I.: Join-semidistributive lattices and convex geometries. Advances in Mathematics 173(1), 1–49 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Caspard, N., Monjardet, B.: The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey. Discrete Applied Mathematics 127(2), 241–269 (2003). Ordinal and Symbolic Data Analysis (OSDA ’98), Univ. of Massachusetts, Amherst, Sept. 28–30, 1998

  4. Ganter, B.: Two basic algorithms in concept analysis. In Kwuida, L., and Sertkaya, B., editors, Formal Concept Analysis, 8th International Conference, ICFCA 2010, Agadir, Morocco, March 15-18, 2010. Proceedings, volume 5986 of Lecture Notes in Computer Science, pages 312–340. Springer (2010)

  5. Ganter, B., Reppe, H.: Base points, non-unit implications, and convex geometries. In: International Conference on Formal Concept Analysis, pp. 210–220. Springer (2007)

  6. Ganter, B., Wille, R.: Formal Concept Analysis - Mathematical Foundations. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  7. Gély, A., Nourine, L.: About the family of closure systems preserving non-unit implications in the guigues-duquenne base. In Missaoui, R., Schmidt, J., editors, Formal Concept Analysis, 4th International Conference, ICFCA 2006, Dresden, Germany, February 13–17, 2006. Proceedings, pp. 191–204. Springer, Berlin (2006)

  8. Grätzer, G.: Lattice Theory: Foundation. Springer Basel (2011)

  9. Grätzer, G., Wehrung, F.: Lattice Theory: Special Topics and Applications (Vol 2), 1st edn. Birkhäuser. 2016 édition (2016)

  10. Guigues, J.L., Duquenne, V.: Familles minimales d’implications informatives résultant d’un tableau de données binaires. Mathématiques et Sciences humaines 95, 5–18 (1986)

    MATH  Google Scholar 

  11. Ennaoui, K., Nourine, L.: Polynomial delay Hybrid algorithms to enumerate candidate keys for a relation (2016)

  12. Maier, D.: Theory of Relational Databases. Financial Times Prentice Hall (1983)

  13. Nation, J.B.: Closure operators and lattice extensions. Order 21(1), 43–48 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yoshikawa, H., Hirai, H., Makino, K.: A representation of antimatroids by horn rules and its application to educational systems. Journal of Mathematical Psychology 77, 82–93 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to thank anonymous referees for their suggestions which have greatly improved the presentation of the paper. They also acknowledge the support received from the Agence Nationale de la Recherche of the French government throught the program Investissements dAvenir (16-IDEX-0001 CAP 20-25) and from Tassili Project TASSILI \(N^{o}\)15 MDU944.

Author information

Authors and Affiliations

  1. Université Clermont Auvergne, CNRS, LIMOS, Clermont-Ferrand, France

    Karima Ennaoui & Lhouari Nourine

  2. Unité de Recherche LaMOS, Faculté des Sciences Exactes, Université de Bejaia, 06000, Bejaia, Algérie

    Khaled Maafa

Authors
  1. Karima Ennaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khaled Maafa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lhouari Nourine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karima Ennaoui.

Additional information

Publisher’s Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ennaoui, K., Maafa, K. & Nourine, L. Representations for the Largest Extension of a Closure System. Order 40, 29–45 (2023). https://doi.org/10.1007/s11083-021-09591-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-021-09591-2

Keywords

Search

Navigation