I am interested in triple cup product operations on the cohomology ring $H^*(Y;\Bbb Z/p^r)$ of 3-manifolds. Trying to extract the algebra, I am led to the following question.
Let's fix a prime power $p^r$, and consider finite abelian groups $A$ with $p^r A = 0$, equipped with a homomorphism $\phi: \Lambda^3 A \to \Bbb Z/p^r$. Consider the monoid of such groups under direct sum. Does this have a nice presentation? Since it probably doesn't, instead consider the Grothendieck group on this monoid. Is that any better?
In particular, is it ever possible that the triple cup product 'simplifies' under direct sum? I don't know what invariants I can extract from $\phi$, much less which ones clearly cannot die from direct sum.
