The duality between the anti-exchange closure operators and the path independent choice operators on a finite set

In this paper, we show that the correspondence discovered by Koshevoy [Math. Soc. Sci. 38 (1999) 35] and Johnson and Dean [An algebraic characterization of path independent choice functions, Third International Meeting of the Society for Social Choice and Welfare, Maastricht, The Netherlands, 1996; Locally complete path independent choice functions and their lattices. Preprint, 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 results concerning the 'ordinal' representations of path independent choice functions from the theory of anti-exchange closure operators. (C) 2001 Elsevier Science B.V. All rights reserved.