Tract associated with a partial field

Core idea

Let $P=(G,R)$ be a partial field. Its associated tract has multiplicative group $G$ and null set

N_P= \left\{\sum_i g_i\in\mathbb N[G]: \sum_i g_i=0\text{ in }R\right\}.

The distinguished element $\epsilon$ of the tract is $-1\in G$ . A morphism of partial fields carries ring-null formal sums to ring-null formal sums and therefore induces a morphism of the associated tracts.

What the construction retains

The tract remembers every finite additive relation among elements of $G$ , including relations whose intermediate binary sums leave $G\cup\{0\}$ . It does not retain the ambient ring $R$ as an object; different ambient presentations can therefore determine the same tract.

References

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