Partial field - Knowlpedia

Definition

A partial field in normalized ambient form is a pair $P=(G,R)$ , where $R$ is a commutative unital ring and $G\leq R^\times$ is a subgroup such that $-1\in G$ and $G$ generates $R$ as a ring. Its underlying pointed set is

\underline P=G\cup\{0\}.

Multiplication is inherited from $R$ , while $a+b$ is defined in $\underline P$ exactly when the ring sum $a+b$ again belongs to $\underline P$ .

Why addition is partial

Multiplication and division by nonzero elements stay inside $G$ , but a ring sum of two selected units can leave $G\cup\{0\}$ . When the sum is defined, it has one value; this differs from a hyperfield, where every pair has a nonempty set of possible sums.

Fields and the regular partial field

Every field $K$ gives the partial field $(K^\times,K)$ , for which addition is defined everywhere and recovers the field. The regular partial field

\mathbb U_0=(\{1,-1\},\mathbb Z)

has only $0,\pm1$ as coefficients; for example $1+(-1)=0$ is defined, while $1+1=2$ is not a coefficient and is therefore undefined.

Associated tract

The tract associated with $P$ records every formal sum of elements of $G$ that vanishes in $R$ , including relations whose intermediate binary sums leave the partial field. Partial fields also have associated pastures, but not every tract or pasture comes from a partial field.

Convention warning

Some definitions omit the requirement that $G$ generate $R$ and identify ambient presentations that produce the same partial field. The generation condition is used here to remove irrelevant ambient ring elements.

References

Charles Semple and Geoff Whittle, “Partial fields and matroid representation,” Advances in Applied Mathematics 17 (1996), 184–208. Institutional record and text. Relevant: the original partial-field framework.
Matthew Baker and Nathan Bowler, “Matroids over partial hyperstructures,” Advances in Mathematics 343 (2019), 821–863. arXiv:1709.09707. Relevant: Definition 2.24 and the associated tract.