Can two logic models with different domains be isomorphic? I'm trying to find out how logic model isomorphism works. 
I have a question. 
If I have models: A: L (language) = {P (predicate)}, M (domain) = {1,2} where P=(1)
B: L = {P}, M = {2,3} where P=(2)
C: L = {P}, M = {1,2,3} where P=(1)

I suppose that A and B models are isomorphic because it's the same thing (1=2 and 2=3)
But I can't figure out whether A and C are isomorphic. They has the same predicate but they hasn't the same domain. 
A: Two isomorphic structures may have different domains, but A and C are not isomorphic. 
Two structures $X$ and $Y$ are isomorphic if there is a bijection $f$ between them which preserves the structure (e.g. if $P(a)$ holds in $X$, then $P(f(a))$ holds in $Y$, etc.). Since A has two elements and C has three, there are no bijections between them at all, let alone structure-preserving ones!
There is an embedding of A into C - if we consider the map $1\mapsto 1, 2\mapsto 2$, this is an injection from A to C, and preserves structure. However, it's not an isomorphism, since it's not a bijection.
Note that "structure-preserving maps" are called homomorphisms - I've used the phrase above since I think that makes things clearer in the beginning.
