Definition
Ordered blue scheme
A scheme locally modeled on spectra of ordered blueprints.
Definition
An ordered blue scheme is an ordered blueprinted space that is locally isomorphic to an affine spectrum of an ordered blueprint .
Why the ordered version is useful
Directed additive relations can encode valuations, hyperaddition, and tropical bend relations. Consequently ordered blue schemes provide a common scheme-theoretic setting for:
- tropicalizations defined by universal valuation problems;
- geometric families of matroids;
- coefficient objects such as hyperfields and pastures through their ordered-blueprint avatars.
Relation to blue schemes
An ordinary blue scheme yields an ordered blue scheme through the fully faithful embedding of blueprints into ordered blueprints on affine charts. General ordered blue schemes need not come from blue schemes because their orders can contain genuinely one-way relations.
The adjective “ordered” modifies the coordinate algebra, not an ordering of the underlying topological space.
References
- Oliver Lorscheid, A unifying approach to tropicalization, §§3 and 7.
- Matthew Baker and Oliver Lorscheid, The moduli space of matroids, Parts I–II.