Let's say we have the structure $(Z, Neigh, Even)$ where $Even(x)$ is true iff $x$ is even and $Neigh(x, y)$ is true iff $x = y + 1$ or $x = y − 1$. I wish to, if possible, construct a function $g$ from $Z$ to itself which is a homomorphism but NOT a strong homomorphism but I am not sure how that would work.
I am using the following definition: Assume that there are two structures A and B with the same language, that is, using the same relations and functions. A mapping h from the domain A of A to the domain B of B is called a homomorphism iff it satisfies the following conditions:
- For every constant symbol $c, h(c^A) = c^B$;
- For every n-ary function symbol f and all $a_1, . . . , a_n ∈ A,h(f^A(a_1, . . . , a_n) = f^B(h(a_1), . . . , h(a_n))$;
- For every n-ary predicate symbol P and all $a_1, . . . , a_n ∈ A, P^A(a_1, . . . , a_n) ⇒ P^B(h(a_1), . . . , h(a_n))$.
A homomorphism h which satisfies in the third item for all predicates P and all $a_1, . . . , a_n$ ∈ A the more restrictive condition $P^A(a_1, . . . , a_n) ⇔ P^B(h(a_1), . . . , h(a_n))$ is called a strong homomorphism.