Definition talk:Isomorphism

Are these all different things? I don't see why, say, an order isomorphism is different from a group isomorphism.

Even if you distinguish mappings that preserve relations and mappings that preserve operations; and order isomorphism is still the same thing as a relation isomorphism.

Not that it matters, it's just a disambiguation page feels slightly out of place here. --Linus44 00:39, 16 June 2011 (CDT)

A good point which we could address.

No, when it comes down to it, they're not different, they're all the same from a high-level perspective. But bear in mind that much of this site has been approached from (and should be accessible to) levels of mathematics which do not take these high-level structures into account.

When investigating elementary results in graph theory, for example, it is useful to have a page that identifies what "isomorphism" means in the limited context of graph theory, without needing to get one's head round what a graph is in the context of category theory (notwithstanding the fact that they are exactly the same anyway, according to the work that's just been posted up).

Maybe there's a case for changing this "disambiguation" page into the same sort of thing that we did for "Continuity" or was it "convergence" (or both) where we used the concept of transcluding more specific results into the more general so as to create a composite page that gives the whole picture at a glance, while allowing for the more specialised and detailed definitions in the various contexts. (**)

Hey, now that would fly. --prime mover 01:29, 16 June 2011 (CDT)

I agree it's important to have the individual definitions for groups, rings etc. so it's accessible without category theory or universal algebra.

I meant that while "isomorphism (abstract algebra)" has a collection of such individual definitions, breaking up relation and order isomorphisms into separate pages seems fairly arbitrary.

Since they're essentially the same thing I thought it'd be tidier to put them all together something like you're suggesting in (**). The notion of disambiguation in this context feels wrong since the concept is essentially unchanged by the context, whereas something like "algebra" has more significantly different meanings.

So I was thinking of a merger of the pages into just "isomorphism" whilst keeping all the individual definitions they contain.

Unrelatedly, should isomorphism (topology) link to homeomorphism? --Linus44 03:16, 16 June 2011 (CDT)

An isomorphism in the context of abstract algebra is a mapping from $S \times S \to S$ as it's a binary operation which satisfies the conditions it does.

An order isomorphism is a mapping $S \to S$ preserving order. As such, it's a different quality of thing. It's a relation isomorphism.

Of course, it's easy to argue (and in the context of set theory this frequently is taken as a basis for thinking) that all mappings $S^n \to S$ are the same sort of thing as each other whatever $n$ may be: 1, 2, ... But from the basis of elementary "abstract algebra", the concepts are usually introduced separately. And you may well want to keep the definition of a "ring isomorphism" etc. on an ordered set separate from an "order isomorphism" as they're used in different contexts.

I rest my case. --prime mover 13:55, 16 June 2011 (CDT)

"... should isomorphism (topology) link to homeomorphism?" I don't know. Presumably they're the same thing, but I haven't encountered the term "isomorphism" in the context of topology, I've only seen "homeomorphism". If we're certain they're the same thing, then yes. The concept was put together by a seagull editor (i.e. flew in one day, made lots of noise, crapped all over everything, flew off again) who didn't explain his terms. --prime mover 13:59, 16 June 2011 (CDT)

Not sure I agree, but I'm happy to leave it as is. Also after reading a little i noticed "order isomorphism" is often used for an order preserving isomorphism of groups that are also posets, so perhaps it is better left separate.

Regarding topological isomorphisms; according to wikipedia the term is used for homeomorphism. May as well have a redirect? --Linus44 10:45, 18 June 2011 (CDT)

Redirect "isomorphism" to "homeomorphism" and add a note to the latter page that the word "isomorphism" is sometimes used to mean this, okay, that would work. I would also recommend that where "isomorphism" is used in the context of topology, it is replaced by "homeomorphism". Why have two terms where one would do? We pick an option and go with it - having two different terms for the same thing can cause confusion. --prime mover 14:56, 18 June 2011 (CDT)

... but now I review it, I see that the usage of isomorphism in the topological context it was meant group isomorphism. So no worries after all. --prime mover 15:36, 18 June 2011 (CDT)