Order

, Volume 26, Issue 1, pp 69–94

# Choice Functions and Extensive Operators

• V. Danilov
• G. Koshevoy
Article

## Abstract

The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider classes of heritage choice functions satisfying conditions M, N, W, and C, or combinations of these conditions. In terms of extensive operators these classes can be considered as generalizations of symmetric, anti-symmetric and transitive binary relations. Among these classes we meet the well-known classes of matroids and convex geometries. Using a ‘topological’ language we discuss these classes of monotone extensive operators (or heritage choice functions) in terms of neighborhood systems. A remarkable inversion on the set of choice functions is introduced. Restricted to the class of heritage choice functions the inversion turns out to be an involution, and under this involution the axiom N is auto-inverse, whereas the axioms W and M change places.

## Keywords

Neighborhood system Pre-topology Matroid Anti-matroid Exchange and anti-exchange conditions Closure operator Direct image

## References

1. 1.
Aigner, M.: Combinatorial Theory. Springer, Berlin (1979)
2. 2.
Aizerman, M.A., Aleskerov, F.T.: Theory of Choice. North Holland, Amsterdam (1995)
3. 3.
Aizerman, M.A., Zavalishin, N.V., Pyatnitsky, Ye.S.: Global functions of sets in the theory of alternative selection. Avtom. Telemeh. 38(3), 11–125 (1977)Google Scholar
4. 4.
Ando, K.: Extreme point axioms for closure spaces. Discrete Math. 303, 3181–3188 (2006)
5. 5.
Caspard, N., Monjardet, B.: The lattice of closure systems, closure operators and implicational systems on a finite set: a survey. Discrete Appl. Math. 127(2), 241–269 (2003)
6. 6.
Danilov, V., Koshevoy, G.: Mathematics of Plott choice functions. Math. Soc. Sci. 49, 245–272 (2005)
7. 7.
Demetrovics, J., Hencsey, G., Libkin, L., Muchnik, I.: On the interaction between closure operations and choice functions with applications to relational databases. Acta Cybern. 10(Nr.3), 129–139 (1992)
8. 8.
Echenique, F.: Counting combinatorial choice rules. Games Econ. Behav. 58, 231–245 (2007)
9. 9.
Edelman, P.H.: Abstract convexity and meet-distribunive lattices. Contemp. Math. 57, 127–149 (1986)
10. 10.
Johnson, M.R., Dean, R.A.: Locally complete path independent choice functions and their lattices. Math. Soc. Sci. 42, 53–87 (2001)
11. 11.
Koshevoy, G.A.: Choice functions and abstract convex geometries. Math. Soc. Sci. 38, 35–44 (1999)
12. 12.
Litvakov, B.M.: Choice mechanisms using graph-multiple structures. Avtom. Telemeh. 9, 145–152 (1980)
13. 13.
Mirkin, B., Muchnik, I.: Induced layered clusters, hereditary mappings, and convex geometries. Appl. Math. Lett. 15, 293–298 (2002)
14. 14.
Monjardet, B., Raderanirina, V.: Lattices of choice functions and consensus problems. Soc. Choice Welf. 23, 349–382 (2004)
15. 15.
Moulin, H.: Choice functions over a finite sets: a summery. Soc. Choice Welf. 2, 147-160 (1985)
16. 16.
Muchnik, I., Shwartser, L.: Maximization of generalized characteristic of functions of monotone systems. Avtom. Telemeh. 51, 1562–1572 (1990)
17. 17.
Nehring, K.: Rational choice and revealed preference without binariness. Soc. Choice Welf. 14, 403–425 (1997)

## Authors and Affiliations

1. 1.Central Institute of Economics and Mathematics of the RASMoscowRussia