Kernel of a Semiring Homomorphism

The kernel of a Semiring Homomorphism f : R → S : 𝑓 → 𝑅 𝑆 f:R\to S is the Semiring Ideal ker ⁡ f ⊆ R kernel 𝑓 𝑅 \ker f\subseteq R consisting of the subset of R 𝑅 R that lies in the Fibre of f 𝑓 f over 0 : S : 0 𝑆 0:S .

When we view f : R → S : 𝑓 → 𝑅 𝑆 f:R\to S as defining a Displayed Semiring R ⇾ S ⇾ 𝑅 𝑆 R\rightarrowtriangle S , then the kernel corresponds directly to the vertical portion of R 𝑅 R .