Tract associated with a partial hyperfield

Core idea

Let $P=(G,R)$ be a partial hyperfield, with $G$ a subgroup of the units of the ambient integral hyperring $R$. Its associated tract has multiplicative group $G$ and null set

The construction uses the ambient hypersum in $R$, not only binary partial sums that remain visible in $G\cup\{0\}$. It therefore records null relations even when every attempted parenthesization passes through elements outside the selected coefficient set.

For an ordinary ambient ring this specializes to the partial-field construction. Taking $G=R^\times$ with $R$ a hyperfield specializes to the hyperfield construction.