Stringent hyperfield

Definition

A hyperfield $F$ is stringent if

a\ne-b\quad\Longrightarrow\quad a\boxplus b \text{ is a singleton}

for every $a,b\in F$ . Thus genuine ambiguity can occur only when adding an element to its hyper-additive inverse.

Examples

Every ordinary field is stringent because all of its sums are singletons. The Krasner and sign hyperfields are stringent. Every valuative hyperfield is stringent as well: each nonzero element is its own additive inverse, unequal inputs have the unique maximum as their sum, and only a tied sum can be multivalued.

References

Nathan Bowler and Ting Su, Classification of doubly distributive skew hyperfields and stringent hypergroups.