The decategorification of an essentially small category $\mathcal C$ is the set $\lvert\mathcal C\rvert$ of isomorphism classes of $\mathcal C$.
If $\mathcal C$ carries additional structure, then so does $\lvert\mathcal C\rvert$. For example, the decategorification of a braided monoidal category is a commutative monoid.
Side Question 1: Is there more we can say about the commutative monoid $\lvert\mathcal C\rvert$ in the above example? Are there any general properties, or does every commutative monoid arise as $\lvert\mathcal C\rvert$ for some braided monoidal category $\mathcal C$?
Let $\mathcal T$ be a triangulated category. Then, by the above example, $\lvert\mathcal T\rvert$ is certainly a commutative monoid with respect to direct sum. This only uses the additive structure on $\mathcal T$, of course.
Main Question: How can we describe the additional structure on the commutative monoid $\lvert\mathcal T\rvert$ induced by the triangulated structure on $\mathcal T$?
Side Question 2: If $\mathcal A$ is an Abelian category, which additional properties does the commutative monoid $\lvert\mathcal A\rvert$ have?