Let $\Sigma$ be a fixed signature, and $\A$ and $\B$ be two structures for $\Sigma$ . Given a homomorphism $f\colon \A \to \B$ , the kernel of $f$ is the relation $\ker(f)$ on $A$ defined by$$ \tuple{a,a'} \in \ker(f) \Iff f(a) = f(a').$$ So defined, the kernel of $f$ is a congruence on $\A$ . If $\Sigma$ 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 $\set{0^\B}$ is a substructure of $\B$ , then the kernel of $f$ is a substructure of $\A$ .