kernel

Let $\mathrm{\Sigma }$ be a fixed signature, and $𝔄$ and $𝔅$ be two structures for $\mathrm{\Sigma }$. Given a homomorphism $f:𝔄\to 𝔅$, the kernel of $f$ is the relation $\ker (f)$ on $A$ defined by

$$⟨a,a^{\prime }⟩\in \ker (f)\iff f(a)=f(a^{\prime }).$$

So defined, the kernel of $f$ is a congruence on $𝔄$. If $\mathrm{\Sigma }$ has a constant symbol 0, then the kernel of $f$ is often defined to be the preimage of $0^{𝔅}$ under $f$. Under this definition, if $\{0^{𝔅}\}$ is a substructure of $𝔅$, then the kernel of $f$ is a substructure of $𝔄$.