homomorphisms and binary algebraic structures What exactly does it mean to say a map is "structure preserving"? And what is the difference between this and a isomorphism?  
f:G -> H. 
f(g1 ∗ g2) = f(g1) ∗′ f(g2) . is the definition of a homomorphism. Which seems to say that you can transfer the binary algebraic structure in G to a binary algebraic structure in H. But intuitively, what does that really mean? I have no taken that many math classes, so if you could keep the examples you may use to as basic and primitive as possible, that would be terrific! 
 A: A structure-preserving map $f:G\rightarrow H$ is one that "doesn't care" where you do the operation. For example, if we had groups $G,H$ with operations $\cdot, *$, then a homomorphism is structure-preserving in the sense that if we have elements $g,g' \in G$ and corresponding $f(g),f(g') \in H$, then it doesn't matter whether we do:


*

*Multiply $g\cdot g'$ (in $G$)

*Take the image of the product, $f(g\cdot g')$
Or the other way:
1'. Take the images $f(g), f(g')$
2'. Multiply the images $f(g)*f(g')$ (in $H$)
Either way, we arrive at the same value: that is, $f(g\cdot g') = f(g)*f(g')$, which is exactly the definition of a homomorphism.
Homomorphisms differ from isomorphisms insofar as the latter requires $f$ to be bijective; that is, that there is also an inverse map $f^{-1}:H \rightarrow G$ that is also a homomorphism.
A: I wanted to illustrate with an example, and what I have to say probably won't fit in a comment (not to detract from OrangeSleipnir's answer).
A very common homomorphism, and one that is used over and over again in algebra is the (additive group) homomorphism:
$\phi: \Bbb Z \to \Bbb Z_n$ given by $\phi(k) = [k]_n$ ($k$ (mod $n$)).
As OrangeSleipnir points out, the fact that $\phi$ "respects addition" means we can add $k$ and $m$ and then reduce mod $n$, or reduce $k$ and $m$ mod $n$ first, and then add the resulting equivalence classes (residues). In symbols:
$[k + m]_n = [k]_n + [m]_n$, that is:
$\phi(k+m) = \phi(k) + \phi(m)$.
However, some information is "lost in translation"; for example, in the integers, $4 + 4 \neq 1 + 2$, but in the integers modulo $5$:
$[4]_5 + [4]_5 = [1]_5 + [2]_5$.
This is because $\phi$ maps both $8$ and $3$ to $[3]_5$.
With "isomorphisms" there isn't any such "condensation" taking place, no information is lost translating from one structure to another. The structure is not only preserved (the "group-ness" or "ring-ness" or "vector space-ness"), but faithfully preserved (we may as well just be changing the symbols we are using for objects and operations).
One way to say this is that isomorphism is an equivalence relation on the "set" (typically a proper class, usually) of all examples of a certain kind of structure, so is a good stand-in for "equality". Structure-preserving, on the other hand, typically isn't "symmetric", we might "condense", but there's often no unique way to "blow back up". This "asymmetry" raises its head in all sorts of situations-for example, it is one of the reasons an "under-determined" system of linear equations often has no "unique solution" (A matrix is one kind of "vector-space structure-preserving map", that is, it is linear-it preserves linear combinations).
Invertible matrices (by the same token) are examples of linear isomorphisms, and they faithfully preserve the important algebraic features of a vector space, such as dimension. This is why a consistent system of $n$ linear equations in $n$ unknowns has a unique solution (no information is lost).
