A projection property for buildings
An object X of a category is said to have the projection property if the only idempotent morphisms f : X x X --> X are the projections. Here a morphism f : X x X --> X is called idempotent if f circle Delta = id for the diagonal map Delta : X --> X x X. There are two motivations for studying the question whether X has the projection property. Firstly, Arrow's 'dictator theorem' states that the only maps of a product XA to X with certain properties are the projections (see Arrow, 1963; Pouzet et al. 1996). Secondly, the projection property is closely related to the fixed point property (see Corominas, 1990). In that paper the projection property has been introduced for posets. It has been studied in a more general setting by Davey et al. (1994) and Pouzet et al. (1996). For a detailed discussion of the projection property, its background and connections with other properties see also the paper by Pouzet (this volume). In this paper we prove that an irreducible building of spherical type and of rank at least 2 has the projection property. Actually, the theorem is more general. It holds not only for the case of a product of two copies of X but for any finite number of copies of X and is thus similar to Arrow's theorem. For a precise statement of the hypotheses see below. By contrast, every reducible building and every building of rank one does not have the projection property. We also give a counterexample concerning the finiteness assumption of the theorem. (C) 1998 Published by Elsevier Science B.V. All rights reserved.
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