Monoid blueprint to semiring blueprint

Core idea

Let $R$ be a commutative semiring. Its multiplicative monoid with zero $R^\bullet$ determines a monoid blueprint

(R^\bullet)^{\mathrm{blpr}}=R^\bullet/\!/\mathcal R_{\min},

while its addition determines the semiring blueprint

R^{\mathrm{blpr}}=R^\bullet/\!/\mathcal R_R.

Since $\mathcal R_{\min}\subseteq\mathcal R_R$ , the identity on $R^\bullet$ induces a canonical blueprint morphism

(R^\bullet)^{\mathrm{blpr}}\longrightarrow R^{\mathrm{blpr}}.

It is the universal morphism from the monoid blueprint that imposes the additive equalities of $R$ . Its source has the free-semiring completion on $R^\bullet$ , whereas its target has semiring completion $R$ ; therefore this morphism is generally not an isomorphism.

References

Oliver Lorscheid, The geometry of blueprints, Part I, §§1.1–1.2.