A projection property and Arrowʹs impossibility theorem Original Research Article

Corominas (1990) introduced the following notion for posets: P is projective if every map F : P × 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). Arrowʹs 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 ⩾ 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.