---
_id: '1624676'
abstract:
- lang: eng
text: '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.'
article_type: original
author:
- first_name: Herbert
full_name: Abels, Herbert
id: '10478'
last_name: Abels
citation:
ama: Abels H. A projection property for buildings. *Discrete Mathematics*.
1998;192(1-3):3-10.
angewandte-chemie: H. Abels, “A projection property for buildings”, *Discrete
Mathematics*, **1998**, *192*, 3-10.
apa: Abels, H. (1998). A projection property for buildings. *Discrete Mathematics*,
*192*(1-3), 3-10. doi:10.1016/s0012-365x(98)00062-4
apa_indent: Abels,
H. (1998). A projection property for buildings. *Discrete Mathematics*,
*192*(1-3), 3-10. doi:10.1016/s0012-365x(98)00062-4

bio1: 'Abels H (1998)

A projection property for buildings.

Discrete Mathematics
192(1-3): 3-10.'
chicago: 'Abels,
Herbert. 1998. “A projection property for buildings”. *Discrete Mathematics*
192 (1-3): 3-10.

'
default: 'Abels H (1998)

*Discrete Mathematics* 192(1-3): 3-10.'
dgps: Abels,
H. (1998). A projection property for buildings. *Discrete Mathematics*,
*192*(1-3), 3-10. Elsevier. doi:10.1016/s0012-365x(98)00062-4.

frontiers: Abels, H. (1998). A projection property for buildings. *Discrete Mathematics*
192, 3-10.
harvard1: Abels, H., 1998. A projection property for buildings. *Discrete Mathematics*,
192(1-3), p 3-10.
ieee: ' H. Abels, “A projection property for buildings”, *Discrete Mathematics*, vol.
192, 1998, pp. 3-10.'
lncs: ' Abels, H.: A projection property for buildings. Discrete Mathematics. 192,
3-10 (1998).'
mla: 'Abels, Herbert. “A projection property for buildings”. *Discrete Mathematics*
192.1-3 (1998): 3-10.'
wels: 'Abels, H. (1998): A projection property for buildings *Discrete Mathematics*,192:(1-3):
3-10.'
date_created: 2010-04-28T13:20:23Z
date_updated: 2019-05-03T12:55:08Z
department:
- _id: '10020'
doi: 10.1016/s0012-365x(98)00062-4
external_id:
isi:
- '000076420800002'
intvolume: ' 192'
isi: 1
issue: 1-3
language:
- iso: eng
page: 3-10
publication: Discrete Mathematics
publication_identifier:
issn:
- 0012-365X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: A projection property for buildings
type: journal_article
user_id: '67994'
volume: 192
year: '1998'
...