Definition of semi-ring homomorphism

Question

I need the definition of semi-ring homomorphism. Thanks in advance!

Answer 1

The definition should depend on your purposes, but I think that this should be the best definition:

$$f(a+b) = f(a) + f(b)$$ $$f(ab) = f(a)f(b)$$ $$f(0) = 0$$ $$f(1) = 1$$

Unlike with rings, you cannot deduce $f(0) = 0$ from the other axioms.

If you want to work with non-unital semirings, you might drop the last axiom. But as far as I can tell, semirings are always defined to be additive monoids, in which case you definitely should not drop $f(0) = 0$.

Answer 2

Just as for rings: $f(a+b)=f(a)+f(b), f(ab)=f(a)f(b)$.

Answer 3

For any sort of algebra, a homomorphism is a function that preserves the structure.

What do you consider to be the structure of a semiring? $0,1,+,\cdot$? Then the definition of homomorphism is any function $R \to S$ such that

• $f(0) = 0$
• $f(1) = 1$
• $f(a + b) = f(a) + f(b)$
• $f(a \cdot b) = f(a) \cdot f(b)$

The same is true for fields, groups, or whatever.

Aside: the definition of group homomorphism is usually presented in a different way. But it turns out that any function between groups that preserves $\cdot$ also preserves ${}^{-1}$ and $1$, so the above general notion also works in the group case.

Another aside: this is part of why people make a big deal about whether the existence of a multiplicative identity in a ring is specified by giving a constant $1$ that satisfies $\forall x: 1x=x1=x$, or by an existential statement like $\exists e \forall x: ex = xe = x$. The two different options lead to two different notions of ring homomorphism.