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 is an integral domain if and only if 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.
@article{ITA_2008__42_1_83_0,
author = {Pouzet, Maurice},
title = {When is the orbit algebra of a group an integral domain ? Proof of a conjecture of P. J. Cameron},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
volume = {42},
year = {2008},
pages = {83-103},
doi = {10.1051/ita:2007054},
mrnumber = {2382545},
zbl = {1146.03015},
language = {en},
url = {http://dml.mathdoc.fr/item/ITA_2008__42_1_83_0}
}
Pouzet, Maurice. When is the orbit algebra of a group an integral domain ? Proof of a conjecture of P. J. Cameron. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) pp. 83-103. doi : 10.1051/ita:2007054. http://gdmltest.u-ga.fr/item/ITA_2008__42_1_83_0/
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