Unique-weak-inverse reflection of an ordered blueprint

Core idea

For an ordered blueprint $B$ , its unique-weak-inverse reflection is an ordered blueprint $B^\pm$ with unique weak inverses together with a morphism

\eta_B:B\longrightarrow B^\pm

such that every morphism $B\to C$ into an ordered blueprint with unique weak inverses factors uniquely through $\eta_B$ .

In categorical terms, $B\mapsto B^\pm$ is left adjoint to the inclusion

\operatorname{OBlpr}^{\pm}\hookrightarrow\operatorname{OBlpr}.

Thus $\operatorname{OBlpr}^{\pm}$ is a reflective full subcategory.

Construction

One first tensors with $\mathbb F_1^\pm$ to adjoin weak inverses. One then identifies any two elements that serve as weak inverses of the same element. Schematically,

B^\pm= \bigl(B\otimes_{\mathbb F_1}\mathbb F_1^\pm\bigr)/\!\sim,

where $a\sim a'$ when some $b$ satisfies both $0\leq a+b$ and $0\leq a'+b$ . The resulting quotient enforces uniqueness.

Terminology and caution

This reflector was historically called pasteurization. The canonical ID retains that established name, but the title uses current terminology. The reflection need not embed $B$ : its unit $\eta_B$ may identify elements. It is also not a passage from a semiring to a ring; weak inverses are order-theoretic.

References

Oliver Lorscheid, Blueprints and tropical scheme theory, Definition 5.6.30 and Exercise 5.6.31.