Hyperrings as ordered blueprints

Core idea

Let $R$ be a Krasner hyperring. Its associated ordered blueprint $R^{\mathrm{oblpr}}$ has underlying monoid $R^\bullet=(R,\cdot)$ , additive semiring generated by that monoid, and order generated by

a\leq b_1+\cdots+b_n \qquad\Longleftrightarrow\qquad a\in b_1\boxplus\cdots\boxplus b_n.

The zero generator is identified with the zero of the semiring.

With the weak morphism convention

f(a\boxplus b)\subseteq f(a)\boxplus f(b),

this construction defines a fully faithful functor from hyperrings to ordered blueprints.

Recovering the hyperaddition

The monomial inequalities recover the original operation:

a\in b\boxplus c \quad\Longleftrightarrow\quad a\leq b+c.

Thus the embedding does not replace a multivalued sum by an ordinary semiring sum. The semiring expression $b+c$ is a formal additive expression, while the order records which monomials lie beneath it.

Scope and convention warning

Not every ordered blueprint arises from a hyperring. Hyperring images satisfy additional monomiality and reversibility properties, as well as unique weak inverses. Some sources use a slightly different but naturally related encoding; full faithfulness should always be read with the stated hyperring-morphism convention.

References

Matthew Baker and Oliver Lorscheid, The moduli space of matroids, §2.8.5.