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

Maurice Pouzet

RAIRO - Theoretical Informatics and Applications (2008)

  • Volume: 42, Issue: 1, page 83-103
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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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 42.1 (2008): 83-103. <http://eudml.org/doc/250356>.

@article{Pouzet2008,
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. },
author = {Pouzet, Maurice},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Relational structures; ages; counting functions; oligomorphic groups; age algebra; Ramsey theorem; integral domain; orbit algebra; permutation group; relational structure; enumeration of finite substructures},
language = {eng},
month = {1},
number = {1},
pages = {83-103},
publisher = {EDP Sciences},
title = {When is the orbit algebra of a group an integral domain ? Proof of a conjecture of P.J. Cameron},
url = {http://eudml.org/doc/250356},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Pouzet, Maurice
TI - When is the orbit algebra of a group an integral domain ? Proof of a conjecture of P.J. Cameron
JO - RAIRO - Theoretical Informatics and Applications
DA - 2008/1//
PB - EDP Sciences
VL - 42
IS - 1
SP - 83
EP - 103
AB - 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.
LA - eng
KW - Relational structures; ages; counting functions; oligomorphic groups; age algebra; Ramsey theorem; integral domain; orbit algebra; permutation group; relational structure; enumeration of finite substructures
UR - http://eudml.org/doc/250356
ER -

References

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