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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
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    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. 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.
    6. 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.
    7. Vladimir Danilov & Gleb Koshevoy & Ernesto Savaglio, 2012. "Orderings of Opportunity Sets," Working Papers 282, ECINEQ, Society for the Study of Economic Inequality.

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