I am confused about the concept of homomorphism, in particular my lecturer asks us to give examples, which I find myself unable to. Could anyone help please?
According to Hodges' Shorter Model Theory, a homomorphic function $f$ between structures $A$ and $B$ that share signature $L$ (ie. same set of relation and function symbols) is one which preserves functions and relation symbols:
1) For each constant in $L$, $f(c^A)=c^B$
2) For each tuple $\bar a$ from $A$ and relation symbol $R$ of $L$, if $\bar a\in R^A$ then $f(\bar a)\in R^B$
3) For each function symbol $F$ of $L$ and tuple $\bar a$, if $\bar a\in R^A$ then $f(F^A(\bar a))=F^B(f(\bar a))$
So I am guessing one simple example would be, if $A$ and $B$ both have domain of all human, and $=$ is the only symbol, then a homomorphic $f$ would map 'Obama' from $A$ to 'Obama' to $B$ (as per 1) regarding constant symbols)?
But then my lecturer asks us to come up with examples of homomorphic functions between the following:
a) from $(\Bbb N,+)$ to $(\Bbb Z,+)$
b) from $(\Bbb N,<)$ to $(\Bbb Z,<)$
I think I have difficulty understanding 2) and 3) - because I am not even sure what am I supposed to answer here.
I would say that a function $f$ is homomorphic if it maps $+/<$ to $+/<$ respectively; but I am sure the answer is not that simple. In particular, I am not sure:
Why do $\Bbb N$ and $\Bbb Z$ make a difference? Does it matter that if domains are $\Bbb N$?
$+$ and $<$ are the only symbols in the signature, so are they the only symbols we need $f$ to be concerned with, or do we also need to worry about other symbols that built on these, e.g. $-$?