Blueprint as an ordered blueprint

Core idea

Let $B=A/\!/\mathcal R$ be a blueprint. Its associated ordered blueprint is

B^{\mathrm{ord}}=(A,B^+,=),

where $B^+=\mathbb N[A]/\mathcal R$ is the semiring completion, $A$ is its distinguished multiplicative generating subset, and the order on $B^+$ is equality.

Equivalently, each additive relation $\sum a_i\equiv\sum b_j$ may be encoded by the two inequalities

\sum a_i\leq\sum b_j \qquad\text{and}\qquad \sum b_j\leq\sum a_i.

Fully faithful embedding

The assignment $B\mapsto B^{\mathrm{ord}}$ defines a fully faithful functor

\mathbf{Blpr}\hookrightarrow\mathbf{OBlpr}.

Indeed, a blueprint morphism induces an order-preserving semiring homomorphism of completions that preserves the distinguished monoids. Conversely, a morphism between equality-ordered objects preserves precisely the additive equalities defining the original blueprints.

References

Oliver Lorscheid, A unifying approach to tropicalization, §2.7.