A projection property and Arrow's impossibility theorem

Corominas (1990) introduced the following notion for posets: P is projective if every map f : P x P --> P which is order-preserving and idempotent is one of the two projections. Since then, extensions of this notion to other structures than posets, as well as maps with n variables, have been considered (Davey et al., 1994; Pouzet et al., 1996; Abels, 1998). Arrow's impossibility theorem (for linear orders) has been rephrased as the projection property of a relational structure made of some equivalence relations on the collection P-(m) of linear orders on an m-element set (m greater than or equal to 3) (Pouzet et al., 1996). We prove a stronger result: the permutahedron P-(m) graph defined by the union of these equivalence relations, is affine projective. (C) 1998 Elsevier Science B.V. All rights reserved.