Semiring as a blueprint

Core idea

Let $R$ be a commutative semiring, and let $R^\bullet$ be its multiplicative monoid with zero. The blueprint associated with $R$ is

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

where

\sum a_i\equiv_{\mathcal R_R}\sum b_j \quad\Longleftrightarrow\quad \sum a_i=\sum b_j\ \text{in }R.

Thus its pre-addition records all and only the formal additive equalities that hold in $R$ .

Its semiring completion is canonically isomorphic to $R$ . Moreover, the construction defines a fully faithful functor

\mathbf{CSRng}\hookrightarrow\mathbf{Blpr},

because morphisms between semiring blueprints are exactly homomorphisms of the original commutative semirings.

References