Ordered blue field - Knowlpedia

Definition

An ordered blue field is a nontrivial ordered blueprint $F$ such that

F^\bullet=F^\times\sqcup\{0\};

that is, every nonzero element of its underlying multiplicative monoid is invertible.

Scope

The term concerns the monomial skeleton $F^\bullet$ . It does not assert that the ambient semiring $F^+$ is a field, nor does it by itself supply additive inverses or unique weak inverses.

Fields, partial fields, and hyperfields have ordered-blue-field avatars after their additive data are encoded in the order. General ordered blue fields form a larger class. Additional conditions define idylls and pastures.

References

Matthew Baker and Oliver Lorscheid, The moduli space of matroids, §§2.6 and 6.4.