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The Duality Between the Anti-Exchange Closure Operators and the Path Independent Choice Operators on a Finite Set


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  • Monjardet, B.
  • Raderanirina, V.


In this paper, we show that the correspondence discovered by Koshevoy (1999) and, Johnson and Dean (1998) between anti-exchange closure operators and path independent choice operators is a duality between two semilattices of such operators. Then we use this duality to obtain old and new results concerning the "ordinal" representations of choice functions from the theory of anti-exchange closure operators.

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Bibliographic Info

Paper provided by Université Panthéon-Sorbonne (Paris 1) in its series Papiers d'Economie Mathématique et Applications with number 1999-68.

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Length: 23 pages
Date of creation: 1999
Date of revision:
Handle: RePEc:fth:pariem:1999-68

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Postal: France; Universite de Paris I - Pantheon- Sorbonne, 12 Place de Pantheon-75005 Paris, France
Phone: + 33 44 07 81 00
Fax: + 33 1 44 07 83 01
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  1. Koshevoy, Gleb A., 1999. "Choice functions and abstract convex geometries," Mathematical Social Sciences, Elsevier, vol. 38(1), pages 35-44, July.
  2. Richard A. Dean & Mark R. Johnson, 2000. "Locally Complete Path Independent Choice Functions and Their Lattices," Econometric Society World Congress 2000 Contributed Papers 0622, Econometric Society.
  3. 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).
  4. Plott, Charles R, 1973. "Path Independence, Rationality, and Social Choice," Econometrica, Econometric Society, vol. 41(6), pages 1075-91, November.
  5. Sen, Amartya K, 1971. "Choice Functions and Revealed Preference," Review of Economic Studies, Wiley Blackwell, vol. 38(115), pages 307-17, July.
  6. Edelman, Paul H., 1997. "A note on voting," Mathematical Social Sciences, Elsevier, vol. 34(1), pages 37-50, August.
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Cited by:
  1. Bernard Monjardet & Raderanirina Vololonirina, 2004. "Lattices of choice functions and consensus problems," Post-Print halshs-00203346, HAL.
  2. Bernard Monjardet, 2007. "Some order dualities in logic, games and choices," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00202326, HAL.
  3. 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.
  4. Gabriela Bordalo & Nathalie Caspard & Bernard Monjardet, 2009. "Going down in (semi)lattices of finite Moore families and convex geometries," Post-Print halshs-00308785, HAL.
  5. Danilov, V. & Koshevoy, G., 2006. "A new characterization of the path independent choice functions," Mathematical Social Sciences, Elsevier, vol. 51(2), pages 238-245, March.
  6. Bernard Monjardet, 2007. "Some order dualities in logic, games and choices," Post-Print halshs-00202326, HAL.
  7. Matthew Ryan, 2010. "Mixture sets on finite domains," Decisions in Economics and Finance, Springer, vol. 33(2), pages 139-147, November.
  8. 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.


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