Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-24T00:56:39.825Z Has data issue: false hasContentIssue false
  • English
  • Français

When is the orbit algebra of a groupan 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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

J. Berstel and C. Retenauer, Les séries rationnelles et leurs langages. Études et recherches en Informatique. Masson, Paris (1984) p. 132.
N. Bourbaki, Éléments de mathématiques, Fasc. XI. Algèbre, Chap. V. Actualités scientifiques et industrielles, Hermann, Paris (1973).
Cameron, P.J., Transitivity of permutation groups on unordered sets. Math. Z. 48 (1976) 127–139. CrossRef
Cameron, P.J., Orbits of permutation groups on unordered sets. II. J. London Math. Soc. 23 (1981) 249–264. CrossRef
P.J. Cameron, Oligomorphic permutation groups. Cambridge University Press, Cambridge (1990).
P.J. Cameron, The algebra of an age, in Model theory of groups and automorphism groups (Blaubeuren, 1995). Cambridge University Press, Cambridge (1997) 126–133.
Cameron, P.J., On an algebra related to orbit-counting. J. Group Theory 1 (1998) 173–179. CrossRef
P.J. Cameron, Sequences realized by oligomorphic permutation groups. J. Integer Seq. 3 Article 00.1.5, 1 HTML document (electronic) (2000).
Cameron, P.J., Some counting problems related to permutation groups. Discrete Math. 225 (2000) 77–92. Formal power series and algebraic combinatorics (Toronto, ON, 1998). CrossRef
P.J. Cameron, Problems on permutation groups, http://www.maths.qmul.ac.uk/~pjc/pgprob.html
R. Diestel, Graph Theory. Springer-Verlag, Heidelberg. Grad. Texts Math. 173 (2005) 431.
R. Fraïssé, Cours de logique mathématique. Tome 1: Relation et formule logique. Gauthier-Villars Éditeur, Paris (1971).
R. Fraïssé, Theory of relations. North-Holland Publishing Co., Amsterdam (2000).
Fraïssé, R. and Pouzet, M., Interprétabilité d'une relation pour une chaîne. C. R. Acad. Sci. Paris Sér. A 272 (1971) 1624–1627.
Gottlieb, D.H., A class of incidence matrices. Proc. Amer. Math. Soc. 17 (1966) 1233–1237. CrossRef
R. Graham, B. Rothschild and J.H. Spencer, Ramsey Theory. John Wiley and Sons, NY (1990).
Higman, G., Ordering by divisibility in abstract algebras. Proc. London Math. Soc. 2 (1952) 326–336. CrossRef
W. Hodges, Model Theory. Cambridge University Press, Cambridge (1993) 772.
Kantor, W.M., On incidence matrices of finite projective and affine spaces. Math. Z. 124 (1972) 315–318. CrossRef
Livingstone, D. and Wagner, A., Transitivity of finite permutation groups on unordered sets. Math. Z. 90 (1965) 393–403. CrossRef
M. Lothaire, Combinatorics on words. Encyclopedia of Mathematics and its Applications 17. Addison-Wesley, Reading, Mass. Reprinted in the Cambridge Mathematical Library, Cambridge University Press, U.K. (1997).
Macpherson, H.D., Growth rates in infinite graphs and permutation groups. Proc. London Math. Soc. 51 (1985) 285–294. CrossRef
Pouzet, M., Application d'une propriété combinatoire des parties d'un ensemble aux groupes et aux relations. Math. Z. 150 (1976) 117–134. CrossRef
M. Pouzet, Sur la théorie des relations. Thèse de doctorat d'État, Université Claude-Bernard, Lyon 1 (1978).
Pouzet, M., Relation minimale pour son âge. Z. Math. Logik Grundlag. Math. 25 (1979) 315–344. CrossRef
Pouzet, M., Application de la notion de relation presque-enchaînable au dénombrement des restrictions finies d'une relation. Z. Math. Logik Grundlag. Math. 27 (1981) 289–332. CrossRef
Pouzet, M., Relation impartible. Dissertationnes 103 (1981) 1–48.
M. Pouzet, The profile of relations. Glob. J. Pure Appl. Math. 2 (2006) 237–272 (Proceedings of the 14th Symposium of the Tunisian Mathematical Society held in Hammamet, March 20–23, 2006).
Pouzet, M. and Sobrani, M., Sandwiches of ages. Ann. Pure Appl. Logic 108 (2001) 295–326. CrossRef
M. Pouzet and N. Thiéry, Some relational structures with polynomial growth and their associated algebras. May 10th (2005), p. 19, presented at FPSAC for the 75 birthday of A. Garsia. arXiv:math/0601256v1 [math.CO]
Radford, D.E., A natural ring basis for the shuffle algebra and an application to group schemes. J. Algebra 58 (1979) 432–454. CrossRef
Ramsey, F.P., On a problem of formal logic. Proc. London Math. Soc. 30 (1930) 264–286. CrossRef