Hyperring homomorphism

Definition

For hyperrings $R$ and $S$ , a hyperring homomorphism in the house weak convention is a function $f:R\to S$ such that

f(0)=0,\qquad f(1)=1,\qquad f(ab)=f(a)f(b),

and

f(a\boxplus_R b)\subseteq f(a)\boxplus_S f(b), \qquad f(A)=\{f(x):x\in A\}.

A strong or strict hyperring homomorphism requires equality in the last display for every $a,b$ .

Why the weak convention matters

For an ordinary ring regarded as a singleton-valued hyperring, the weak condition says

f(a+b)\in f(a)\boxplus f(b).

This permits valuations and quotient maps into genuinely multivalued hyperfields. Strong preservation would require the target hyper-sum to be the singleton image of $a+b$ , excluding these central examples.

Relations between the notions

Every strong homomorphism is weak, but not conversely. When both source and target have singleton-valued addition, weak and strong preservation coincide with the usual additive equality. An isomorphism of hyperrings is a bijection whose map and inverse are weak homomorphisms; it is then strong.

Convention warning

The adjectives weak and strong in matroid theory over hyperfields refer to different axiom systems and are not the morphism distinction above. Sources also vary on whether the unqualified word “homomorphism” means weak or strict, so the convention must be checked.

References

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