Pre-addition on a monoid

Let $A$ be a commutative monoid with zero, and let $\mathbb N[A]$ denote the semiring of finite formal sums of elements of $A$ . A pre-addition on $A$ is an equivalence relation $\mathcal R$ on $\mathbb N[A]$ , written

\sum a_i\equiv\sum b_j,

with the following properties:

relations may be added and multiplied termwise;
the empty sum is equivalent to the one-term sum $0$ ;
if the one-term sums $a$ and $b$ are equivalent, then $a=b$ in $A$ .

Explicitly, if $\sum a_i\equiv\sum b_j$ and $\sum c_k\equiv\sum d_\ell$ , then

\sum a_i+\sum c_k\equiv\sum b_j+\sum d_\ell

and

\sum_{i,k}a_ic_k\equiv\sum_{j,\ell}b_jd_\ell.

The third condition says that the pre-addition is proper: it does not identify distinct elements of the underlying monoid.

Role in a blueprint

A blueprint $A/\!/\mathcal R$ records the multiplication of $A$ together with the additive relations in $\mathcal R$ . Quotienting all formal sums by $\mathcal R$ produces its semiring completion.

References

Oliver Lorscheid, The geometry of blueprints, Part I: Algebraic background and scheme theory, §1.1.