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

Published online by Cambridge University Press:  18 January 2008

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

Keywords

Relational structuresagescounting functionsoligomorphic groupsage algebraRamsey theoremintegral domain
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 42 , Issue 1: A nonstandard spirit among computer scientists:a tribute to Serge Grigorieff at the occasion of his 60th birthday , January 2008 , pp. 83 - 103
Copyright
© EDP Sciences, 2007

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