Field embeddings into semifields and hyperfields

Statement

There are fully faithful embeddings

\mathbf{Field}\hookrightarrow\mathbf{SemiField}, \qquad \mathbf{Field}\hookrightarrow\mathbf{HyperField}.

The first sends a field to its underlying semifield. The second regards each ordinary sum $a+b$ as the singleton hyper-sum $\{a+b\}$ , producing a hyperfield.

The essential image of the first embedding consists of the semifields whose additive commutative monoid is an abelian group. The essential image of the second consists of the hyperfields whose every hyper-sum is a singleton.

Full faithfulness

A unit-preserving semiring homomorphism between fields preserves additive inverses, because

f(a)+f(-a)=f(0)=0.

It is therefore a field homomorphism. A weak hyperfield homomorphism between singleton-addition hyperfields satisfies

\{f(a+b)\}\subseteq\{f(a)+f(b)\},

so it preserves addition by equality and is likewise a field homomorphism.

Scope

This theorem identifies two copies of the category of fields; it does not identify all semifields with all hyperfields. Semifield addition is a single-valued operation without required additive inverses, whereas hyperfield addition is nonempty-set-valued and satisfies the inverse and reversibility axioms of a canonical hypergroup.

References

Matthew Baker and Nathan Bowler, “Matroids over hyperfields,” 2017. arXiv:1601.01204. Relevant: §2.
Jaiung Jun, “Algebraic Geometry Over Hyperrings,” Advances in Mathematics 323 (2018), 142–192. arXiv:1512.04837. Relevant: §2.