Isomorphism

Remarks

The condition on $h^{-1}$ means that $h$ also reflects relations: $(a_1, \dots, a_n) \in R^{\mathbf{A}}$ if and only if $(h(a_1), \dots, h(a_n)) \in R^{\mathbf{B}}$. For a signature without relation symbols, the inverse of a bijective homomorphism is automatically a homomorphism, so an isomorphism is the same as a bijective homomorphism.

Example: the map $x \mapsto e^x$ is an isomorphism from $(\R, +, 0)$ to $(\R_{>0}, \cdot, 1)$, because it is a bijection with $e^{x + y} = e^x e^y$ and $e^0 = 1$; its inverse is $\ln$.

Non-example: let $A = \set{1, 2}$ with the empty binary relation and $B = \set{1, 2}$ with the relation $\le$. The identity map is a bijective homomorphism from $A$ to $B$, but its inverse does not preserve the relation, since $1 \le 2$ in $B$ while $(1, 2)$ is not related in $A$.