  The profile of a relational structure $R$ is the function $\varphi_R$ which
counts for every integer $n$ the number, possibly infinite, $\varphi_R(n)$ of
substructures of $R$ induced on the $n$-element subsets, isomorphic
substructures being identified. If $\varphi_R$ takes only finite values, this
is the Hilbert function of a graded algebra associated with $R$, the age
algebra $A(R)$, introduced by P.~J.~Cameron.
  In a previous paper, we studied the relationship between the properties of a
relational structure and those of their algebra, particularly when the
relational structure $R$ admits a finite monomorphic decomposition. This
setting still encompasses well-studied graded commutative algebras like
invariant rings of finite permutation groups, or the rings of quasi-symmetric
polynomials.
  In this paper, we investigate how far the well know algebraic properties of
those rings extend to age algebras. The main result is a combinatorial
characterization of when the age algebra is finitely generated. In the special
case of tournaments, we show that the age algebra is finitely generated if and
only if the profile is bounded. We explore the Cohen-Macaulay property in the
special case of invariants of permutation groupoids. Finally, we exhibit
sufficient conditions on the relational structure that make naturally the age
algebra into a Hopf algebra.
