Ordered blueprint - Knowlpedia

Definition

An ordered blueprint is a triple

B=(B^\bullet,B^+,\leq)

in which $(B^+,\leq)$ is an ordered semiring and $B^\bullet\subseteq B^+$ is a multiplicatively closed subset containing $0$ and $1$ that generates $B^+$ as a semiring. A morphism is an order-preserving semiring homomorphism $f^+:B^+\to C^+$ satisfying $f^+(B^\bullet)\subseteq C^\bullet$ .

Presentation form

Equivalently, begin with a commutative monoid $A$ with zero and impose a compatible partial order on its free semiring $\mathbb N[A]$ , possibly identifying elements first. One writes

A/\!/\langle{\text{generating inequalities}}\rangle.

The monoid $B^\bullet$ records the designated monomials; the ordered semiring $B^+$ records their finite sums and additive inequalities.

Relation to blueprints

An ordinary blueprint gives an ordered blueprint by reading each additive equality as inequalities in both directions. The converse fails in general because an inequality need not be symmetric. This extra directionality is what allows one framework to contain semiring orders, hyperaddition, and bend relations without identifying them.

Important subcategories

Objects with unique weak inverses provide the coefficient setting used for much of matroid geometry.
Hyperrings embed fully faithfully after hyperaddition is encoded by monomial inequalities.
Affine spectra glue to ordered blue schemes.

References

Oliver Lorscheid, A unifying approach to tropicalization, §§2–3.
Matthew Baker and Oliver Lorscheid, The moduli space of matroids, §2.6.