7
$\begingroup$

I was reading about isomorphisms and homomorphisms on general structures, and first came across the definition of an injective homomorphism, or an embedding. This made me curious, is it possible for two structures $A$ and $B$ to be embeddable in each other, yet no isomorphism exists between them?

After some looking around, I let the structures be $A=\mathbb{R}$ and $B=[-1,1]$ with $f\colon [-1,1]\to\mathbb{R}\colon r\mapsto r$ and $g\colon\mathbb{R}\to [-1,1]\colon r\mapsto \frac{2}{\pi}\arctan(r)$. If I refrain from defining any relations, functions, or distinguished elements in the universes of $A$ and $B$, then it is vacuously true that $f$ and $g$ are homomorphisms. Also, $[-1,1]$ and $\mathbb{R}$ would not be isomorphic since $[-1,1]$ has a maximum and minimum element. (Or would this require me to define $\lt$ on $[-1,1]$?).

Are there some other structures, even contrived ones, where such embeddings $f$ and $g$ exist, but $A$ and $B$ are still not isomorphic?

  • 0
    $g$ is not injective?2010-09-20
  • 0
    Ouch, my mistake. I was thinking of the tangent function not being injective when I wrote that.2010-09-20
  • 0
    But this is not a problem for your counter-example: it is fine. $f$ and $g$ are indeed injective and continuous, but clearly $[-1,1]$ and $\mathbb{R}$ couldn't be isomorhpic as topological spaces (homeomorphic) -I would've said just because one is compact and the other one isn't.2010-09-20

2 Answers 2

10

Yes. This was discussed on MO and many examples were given.

See https://mathoverflow.net/questions/1058/when-does-cantor-bernstein-hold and the links thereon.

  • 0
    Oops, I apologize if this is considered a duplicate.2010-09-20
  • 2
    @yunone, I do not consider this a duplicate, but it does not hurt to look in both places before asking (and reflect a bit on the saneness of the current situation...)2010-09-20
4

There are unenlightening examples if you don't ask for the image of the embedding to be large.

For example, let A be a square and B an annulus, considered as topological spaces. A small copy of each one can be placed inside the other but they are not isomorphic in the topological category.

Bidirectional embedding is an equivalence relation, as is bidirectional dense (epimorphic) embedding, so at least linguistically the more natural problem in a given category is to ask what is the same about A and B if they are equivalent in this sense.