Ordered blueprint with unique weak inverses

Definition

An ordered blueprint $B$ has unique weak inverses if, for every $a\in B^\bullet$ , there is a unique $b\in B^\bullet$ such that

0\leq a+b.

The element $b$ is the weak inverse of $a$ .

Equivalently, $B$ is an $\mathbb F_1^{\pm}$ -algebra with unique weak inverses, where

\mathbb F_1^{\pm}=\{0,1,\epsilon\}/\!/\langle 0\leq 1+\epsilon\rangle \quad\text{and}\quad \epsilon^2=1.

The weak inverse of $a$ is then $\epsilon a$ .

Terminology

Earlier versions of the ordered-blueprint literature called these objects pasteurized ordered blueprints. Current Baker–Lorscheid usage prefers “ordered blueprints with unique weak inverses” (or $\mathbb F_1^\pm$ -algebras with unique weak inverses). The legacy phrase is retained here only as an alias for search and older citations.

The word “weak” matters: $0\leq a+\epsilon a$ is an order relation, not necessarily an equality in the ambient semiring $B^+$ . Consequently $\epsilon$ should not automatically be written as an additive inverse $-1$ .

Structural role

Hyperrings and tracts have associated ordered blueprints with unique weak inverses. The inclusion of this full subcategory into all ordered blueprints is reflective; its reflector is the unique-weak-inverse reflection.

References

Matthew Baker and Oliver Lorscheid, The moduli space of matroids, §§2.6, 2.9.
Oliver Lorscheid, Blueprints and tropical scheme theory, §5.6.