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

Pouzet, Maurice

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 83-103.

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.

Classification : 03C13, 03C52, 05A16, 05C30, 20B27
Mots clés : relational structures, ages, counting functions, oligomorphic groups, age algebra, Ramsey theorem, integral domain

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