  The profile of a relational structure R is the function phi_R which counts
for every integer n the number, possibly infinite, phi_R(n) of substructures of
R induced on the n-element subsets, isomorphic substructures being identified.
Several graded algebras can be associated with R in such a way that the profile
of R is simply the Hilbert function. An example of such graded algebra is the
age algebra introduced by P.~J.~Cameron.
  In this paper, we give a closer look at this association, particularly when
the relational structure R decomposes into finitely many monomorphic
components. In this case, several well-studied graded commutative algebras
(e.g. the invariant ring of a finite permutation group, the ring of
quasi-symmetric polynomials) are isomorphic to some age algebras. Also, phi_R
is a quasi-polynomial, this supporting the conjecture that, with mild
assumptions on R, phi_R is a quasi-polynomial when it is bounded by some
polynomial.
