According to WP article on Hall algebras one counts the number of abelian subgroups in abelian group with fixed type of subgroup, group, quotient.
Two things are claimed:
1) These numbers are polynomials in "p"
2) Using these numbers one naturally defines the algebra structure on the isomorphism classes of abelian groups which appears to be associative and commutative.
Question: What happens if we consider all p-groups, not just abelian one ? Will same/similar claims be true ?
The natural context for the question seems to me some categories with finite number of exact triples A->B->C for fixed A,B,C. So natural generalization - what the properties of the categories for which we can define associative algebra ? commutative algebra ?