Up to, equivalence and mod. (Read at your peril)

[Ben]

Well, I agree with all of this. So, although my post was merely intended to explain the various sorts of "equality" that sets may have up to any equivalence relation, I thank you for your clarification about isomorphism.

Incidentally: Every text I have ever seen invariably introduces isomorphism first, just as you explained it, and then subsequently goes on to explain homomorphism.

This seems perverse to me; surely a better approach would be first to describe homomorphism as a mapping that respects operation(s) and identity(s), and then simply state that this mapping will be an isomorphism iff it is a bijection?

[Nootropic]

Well it's certainly of interest to note that on the set of structures you define the notion of an isomorphism on, you can actually define an equivalence relation by saying A ~ B iff A is isomorphic to B. Though this isn't much of an interest, it's kind of a special case of the notion of an equivalence relation.

As for textbooks, most textbooks I've read (if I remember correctly...) introduce the idea of a homomorphism before isomorphism, but honestly, it's not that large of a pedagogical disadvantage (and may even, in fact, be an advantage to some). The reasoning behind introducing an isomorphism first is that you can use easy examples to demonstrate this whole idea of "structure-preserving". For example, you might want to see show that Z/12Z is isomorphic to Z/4Z x Z/3Z (as groups). Now an easy way of going about doing this is comparing operation tables and in this way you sort of define a "map", which just so happens to be a bijection as well, but the important part is for your students (or whomever you are teaching) is to see the way in which the operation (in this case, addition) is "essentially" the same. Now it might be a little more difficult to use Z/12Z and Z/6Z and demonstrate a homomorphism from Z/6Z into Z/12Z by means of a group table. I find that this sort of elementary comparison between simple algebraic objects tends to give students a better initial grasp when they don't have to worry about things differing in size. Either way though, it's not nearly as important as something like whether or not you should introduce groups or rings first.

[Qfwfq]

Originally Posted by Nootropic

So in short, what I'm saying is, one must specify what structure you are defining an isomorphism on.

Certainly. I was meaning to say a few words on this and give a very simple yet cute example where the operations in each structure are quite heterogeneous; the exponential is an isomorphism between with ordinary addition and the positive reals with ordinary multiplication. Nobody in their right mind would say these two structures are equal.

Originally Posted by Ben

Every text I have ever seen invariably introduces isomorphism first, just as you explained it, and then subsequently goes on to explain homomorphism.

I quite agree with you in finding this perverse, my courses gave endomorphism, isomorphism and automorphism as specific cases of homomorphism.

Originally Posted by Nootropic

Well it's certainly of interest to note that on the set of structures you define the notion of an isomorphism on, you can actually define an equivalence relation by saying A ~ B iff A is isomorphic to B. Though this isn't much of an interest, it's kind of a special case of the notion of an equivalence relation.

This is kinda what Ben did, just to give an example of equivalence relation.

I also meant to say that an equivalence class of isomorphic structures are sometimes said to be an abstract structure, of which each class member is said to be a concrete representation.