Idyll as an ordered blue field

Statement

Under the fully faithful realization of bands as ordered blueprints, an idyll $B$ becomes a purely positive ordered blue field with unique weak inverses and ambient semiring

B^+=\mathbb N[B^\times].

Its order is generated by

0\leq\sum a_i \quad\Longleftrightarrow\quad \sum a_i\in N_B.

Conversely, an ordered blue field with these properties determines an idyll: take its pointed multiplicative monoid as $B$ and declare a formal sum null exactly when $0\leq\sum a_i$ . These constructions are inverse on objects and morphisms.

References

Matthew Baker, Tong Jin, and Oliver Lorscheid, New building blocks for $\mathbb F_1$ -geometry: bands and band schemes, §1.2; Matthew Baker and Oliver Lorscheid, The moduli space of matroids, §2.