Tract associated with a hyperfield

Core idea

Let $F$ be a hyperfield. Its associated tract has multiplicative group $F^\times$ and null set

N_F= \left\{\sum_i a_i\in\mathbb N[F^\times]: 0\in\mathop{\boxplus}_i a_i\right\}.

The unique hyper-additive inverse of $1$ is the tract's distinguished element $\epsilon=-1$ .

A weak hyperfield homomorphism preserves null hypersums, so restriction to the nonzero multiplicative groups induces a tract morphism.

Scope

This construction forgets the individual values of a non-null hyper-sum and retains its null relations. Every hyperfield therefore determines a tract, but an arbitrary tract need not carry a binary hyperaddition from which its null set can be recovered.

References

Matthew Baker and Nathan Bowler, “Matroids over partial hyperstructures,” Advances in Mathematics 343 (2019), 821–863. arXiv:1709.09707.