IDEAS home Printed from https://ideas.repec.org/p/fth/pariem/2000.120.html
   My bibliography  Save this paper

The Lattice of Closure Systems, Closure Operators and Implicational Systems on a Finite Set : A Survey

Author

Listed:
  • Caspard, N.
  • Monjardet, B.

Abstract

We present a survey of properties of the lattice of closure systems (families of subsets of a set S containing S and closed by set intersection) on a finite set S with proofs of the more significant results. In particular, we prove that this lattice is atomistic and lower bounded and that there exists a canonical basis allowing to represent any closure system by "implicational" closure systems. The notion of closure system has many cryptomorphic versions, especially the notions of closure operator and of (full) implicational system, occuring in many fields of pure or applied mathematics and of computer science.

Suggested Citation

  • Caspard, N. & Monjardet, B., 2000. "The Lattice of Closure Systems, Closure Operators and Implicational Systems on a Finite Set : A Survey," Papiers d'Economie Mathématique et Applications 2000.120, Université Panthéon-Sorbonne (Paris 1).
    • Handle: RePEc:fth:pariem:2000.120
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Gabriela Bordalo & Nathalie Caspard & Bernard Monjardet, 2009. "Going down in (semi)lattices of finite Moore families and convex geometries," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00308785, HAL.
    2. V. Danilov & G. Koshevoy & E. Savaglio, 2015. "Hyper-relations, choice functions, and orderings of opportunity sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(1), pages 51-69, June.
    3. Monjardet, Bernard & Raderanirina, Vololonirina, 2001. "The duality between the anti-exchange closure operators and the path independent choice operators on a finite set," Mathematical Social Sciences, Elsevier, vol. 41(2), pages 131-150, March.
    4. Bernard Monjardet, 2007. "Some Order Dualities In Logic, Games And Choices," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 9(01), pages 1-12.
    5. Vladimir Danilov & Gleb Koshevoy & Ernesto Savaglio, 2012. "Orderings of Opportunity Sets," Working Papers 282, ECINEQ, Society for the Study of Economic Inequality.
    6. Monjardet, Bernard, 2003. "The presence of lattice theory in discrete problems of mathematical social sciences. Why," Mathematical Social Sciences, Elsevier, vol. 46(2), pages 103-144, October.
    7. Jean Diatta, 2009. "On critical sets of a finite Moore family," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 3(3), pages 291-304, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:fth:pariem:2000.120. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thomas Krichel (email available below). General contact details of provider: https://edirc.repec.org/data/cerp1fr.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.