@MISC{Cameron_thealgebra, author = {Peter J. Cameron}, title = {The Algebra of an Age}, year = {} }

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Abstract

Associated with any oligomorphic permutation group G, there is a graded algebra A G such that the dimension of its nth homogeneous component is equal to the number of G-orbits on n-sets. I show that the algebra is a polynomial algebra (free commutative associative algebra) in some cases, and pose some questions about transitive extensions. 1 The algebra Let\Omega be an infinite set. Let \Gamma\Omega n \Delta denote the set of n-element subsets of \Omega\Gamma V n the vector space of functions from \Gamma\Omega n \Delta to Q. Set A = L n0 V n , with multiplication defined as follows: for f 2 V n , g 2 Vm , and X 2 \Gamma\Omega n+m \Delta , (fg)(X) = X Y 2( X n ) f(Y )g(X n Y ): This is the reduced incidence algebra of the poset of finite subsets of\Omega (Rota [13]). It is a commutative and associative algebra with identity, but is far from an integral domain: any function with finite support is nilpotent. Now, if G is any permutation group on \Omega\Gam...