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Issue RAIRO-Theor. Inf. Appl.
Volume 42, Number 1, January-March 2008
A nonstandard spirit among computer scientists:
a tribute to Serge Grigorieff at the occasion of his 60th birthday
Page(s) 83 - 103
DOI 10.1051/ita:2007054
Published online 18 January 2008

RAIRO-Theor. Inf. Appl. 42, 83-103 (2008)
DOI: 10.1051/ita:2007054

When is the orbit algebra of a group an integral domain? Proof of a conjecture of P.J. Cameron

Maurice Pouzet

ICJ, Mathématiques, Université Claude-Bernard - Lyon 1, 43, Bd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France; pouzet@univ-lyon1.fr


(Published online: 18 January 2008)

Abstract
Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure R is an integral domain if and only if R is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra.


Mathematics Subject Classification. 03C13, 03C52, 05A16, 05C30, 20B27

Key words: Relational structures -- ages -- counting functions -- oligomorphic groups -- age algebra -- Ramsey theorem -- integral domain


© EDP Sciences 2007


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