Semiring homomorphism

Definition

For semirings $S$ and $T$ , a semiring homomorphism is a function $f:S\to T$ satisfying

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

for all $a,b\in S$ . In particular, semiring homomorphisms are unit-preserving in the house convention.

Categorical role

Identity functions and composites are semiring homomorphisms, so semirings and these maps form a category. Restricting to commutative semirings gives the category used in the tropical-algebra pages.

Convention warning

Some authors permit maps with $f(1)\ne1$ , especially when studying ideals or nonunital semirings. Such a map is not a semiring homomorphism here unless it is explicitly called nonunital. This differs from a weak hyperring homomorphism: ordinary semiring addition is single-valued, so its preservation is an equality rather than an inclusion.

References

Jonathan S. Golan, Semirings and their Applications, Kluwer, 1999. Publisher DOI record. Relevant: semiring morphisms in Chapter 1.