Commutative monoid with zero as a blueprint

Core idea

Let $A$ be a commutative monoid with zero. Its monoid blueprint is

where $\mathcal R_{\min}$ is the smallest pre-addition on $A$: apart from the blueprint axioms, it imposes no additive relations.

This construction defines a fully faithful functor

A blueprint morphism between two such objects has no additional additive relations to check, so it is exactly a multiplicative map preserving $0$ and $1$.

The semiring completion of $A^{\mathrm{blpr}}$ is the free commutative semiring generated by the monoid $A$, with the monoid zero identified with the additive zero. It is not an addition law on $A$.