Definition 2.1 (Homomorphism). Let $M, N$ be $L$ -structures. A function $h : M \to N$ is an $L$ -homomorphism if:

(i) For an $n$ -ary function symbol $f$ , and $a_{1}, \dots, a_{n} \in M$ we have
$h (f^{M} (a_{1}, \dots, a_{n})) = f^{N} (h (a_{1}), \dots, h (a_{n})) .$
(ii) For an $n$ -ary relation symbol $R$ , and $a_{1}, \dots, a_{n} \in M$ we have
$(a_{1}, \dots, a_{n}) \in R^{m} iff (h (a_{1}), \dots, h (a_{n})) \in R^{N} .$
(iii) For any constant symbol $c$ , $h (c^{M}) = c^{N}$ .

We write $h : M \to N$ if $h$ is an $L$ -homomorphism.

If $h$ is also injective then this is called an $L$ -embedding.

If $h$ is also bijective then this is called an $L$ -isomorphism.