TITLE: Isomorphisms: preserve structure, operation, or order?
QUESTION [8 upvotes]: Everyone always says that isomorphisms preserve structure... but given the (multiple) definitions of isomorphism, I fail to see how the definitions equate with the intuitive meaning, which is that two sets are "basically the same if you ignore naming and notation".
Here are the different definitions I've come across:
Order Isomorphism
Let $A$ be a (totally) ordered set with ordering $\le$ and $B$ be a (totally) ordered set with ordering $\preceq$. An isomorphism of $A$ onto $B$ is a bijection $f:A\mapsto B$ that satisfies
$$a \le b \iff f(a) \preceq f(b)$$
for all $a,b \in A$.
Group Isomorphism
Let $A$ be a group with operation $\ast$ and $B$ be a group with operation $\#$. An isomorphism of $A$ onto $B$ is a bijection $f:A\mapsto B$ that satisfies
$$a \ast b = c \iff f(a) \# f(b) = f(c)$$
rather, put simply,
$$f(a\ast b) = f(a) \# f(b)$$
for all $a,b,c \in A$.
Field Isomorphism
Let $A$ be a field with operations $\#$ and $\ast$ and $B$ be a field with operations $+$ and $\times$. An isomorphism of $A$ onto $B$ is a bijection $f:A\mapsto B$ that satisfies
$$f(a \# b) = f(a) + F(b)$$
and
$$f(a \ast b) = f(a) \times F(b)$$
for all $a,b,c \in A$.
Homomorphism
The same as an isomorphism but not necessarily a bijection.
An example of why this is confusing:
It is technically not the case that $\mathbb{N}\subseteq\mathbb{Z}$. For example, in the Natural Numbers, $0$ is the empty set $\varnothing$, which has no elements, however in the integers, $0$ is the equivalence class $[(0,0)]$ of ordered pairs whose components are natural numbers that are equal, and it has infinitely many elements. These sets are not by any means equal, but for all intents and purposes, we consider $\mathbb{N}$ to be a subset of $\mathbb{Z}$, because the set of non-negative integers is "basically the same as" the set of natural numbers. That is, the natural numbers have "isomorphic copies" in the integers. The question is, which type of isomorphism?
Which of these definitions of isomorphism is "correct", and how does it equate to the intuitive meaning that the two sets are "basically the same"? Furthermore, what's the point of homomorphism, why is it useful, and how is its intuitive meaning similar or different from the intuitive meaning of isomorphism?

REPLY [6 votes]: Each is a correct example of an isomorphism, though they exist within different categories. The idea that isomorphisms between two objects mean the objects are "the same" up to some kind of scheme for renaming the internal elements is a legitimate, albeit limited, interpretation. Suppose we collect "all of the things that could be said" about elements of a group $G$, for example, and use an isomorphism $f$ as a way of renaming the elements of $G$: if we take any of our collected true statements about $G$ and replaced every instance of specific $g\in G$ with $f(g)$s, we would end up with a true statement in the target object. The converse also holds: any true statement involving the elements of the target object translates into a true statement about the elements of the source object. On this view, two structures are "the same" when everything that can be said about them (or their constituent elements) are precisely one and the same (up to naming, of course).
It is kind of like changing the names of characters or places in a story. The plot structure, and indeed various literary features of the story beyond the plot itself, will stay the same. In this way we can imagine it is "the same" story. On the flipside, category theory allows for loftier abstractions of structures: objects do not even need to be comprised of elements for us to do cateogry theory with them. Therefore our interpretation of isomorphisms is restricted in scope to familiar cases where we have sets, and additional algebraic and relational structure defined on them, as our objects.
For many intents and purposes, a morphism can be thought of as putting one structure inside of another structure. It may be the case that we fold up and compress the source object when we put it inside the target object, and it may also be the case that our placement of source inside target does not fill up the entire target object but only a portion of it. This is where we have injectivity and surjectivity come into play. An isomorphism occurs when our placement is a perfect overlay of source onto target: no folding up was necessary, and we fill up all of the room in the target.
To be specific: a failure of injectivity ("folding-up," or "compression") occurs when distinct elements of the source object are sent to a single element in the target object, and a failure of surjectivity occurs when there are elements of the target object with no corresponding element in the source.
This is just some of my personal intuition on the topic.