Partial hyperfield

A partial field has single-valued addition that may be undefined. A hyperfield has nonempty multivalued addition defined for every pair. A partial hyperfield permits both phenomena at once: a visible sum may contain several values or no values.

When the ambient hyperring is an ordinary ring, this construction recovers a partial field provided $R$ is generated as a ring by $G$; equivalently, one may first replace $R$ by the subring generated by $G$. Without this generation condition, the ambient ring can contain redundant elements and the pair is not a partial field in conventions that require generation. When $R$ is a hyperfield and $G=R^\times$, the construction recovers the whole hyperfield.

The tract associated with $P$ records the ambient null hypersums in $R$, even when intermediate sums do not stay inside $\underline P$. Partial hyperfields therefore encompass the tract constructions for both partial fields and hyperfields, but tracts are more general because a tract need not carry any binary partial hyperaddition.

The ambient presentation uses a hyperdomain and a designated subgroup of its units. Requiring $-1\in G$ makes the selected coefficients closed under hypernegation. One may replace $R$ by the subhyperring generated by $G$ to remove redundant ambient elements; some conventions build this normalization into the definition and others do not.

1. Matthew Baker and Nathan Bowler, “Matroids over partial hyperstructures,” Advances in Mathematics 343 (2019), 821–863. arXiv:1709.09707. Relevant: §2.6, partial hyperfields and their associated tracts.
2. Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: hyperfield null sums and the relationship to tract axioms.