Tract - Knowlpedia

A tract is a pair $F=(G,N_F)$ , where $G$ is an abelian group and $N_F\subseteq\mathbb N[G]$ is a set of formal sums, called the null set, satisfying:

the zero formal sum belongs to $N_F$ ;
the one-term sum $1$ does not belong to $N_F$ ;
there is a unique $\epsilon\in G$ with $1+\epsilon\in N_F$ ;
$N_F$ is stable under multiplication by every $g\in G$ .

The underlying pointed multiplicative set is $F=G\sqcup\{0\}$ , with $F^\times=G$ . A morphism of tracts is a group homomorphism $G\to G'$ whose linear extension $\mathbb N[G]\to\mathbb N[G']$ carries $N_F$ into $N_{F'}$ .

What a tract remembers

The relation $\sum g_i\in N_F$ means that the formal sum is declared to be zero. A tract need not provide a binary addition, even a multivalued one. It retains exactly the null relations needed to state orthogonality and Grassmann–Plücker relations for matroids with coefficients.

The distinguished $\epsilon$ behaves as a formal $-1$ : one proves $\epsilon^2=1$ , and if $x+y\in N_F$ then $y=\epsilon x$ .

Examples and inclusions

Fields, partial fields, hyperfields, and partial hyperfields determine tracts through their respective partial-field, hyperfield, and partial-hyperfield constructions. These constructions do not imply that every tract is a hyperfield. Pastures form a structured subcategory related to three-term null relations, and a tract has an associated ordered blueprint.

References

Matthew Baker and Nathan Bowler, Matroids over partial hyperstructures, §1.