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- Thread starter amcavoy
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Icebreaker

It's not an isomorphism because an isomorphism is a function between two rings that preserves the binary operations of those rings, on top of which the function is bijective.

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Icebreaker said:

It's not an isomorphism because an isomorphism is a function between two rings that preserves the binary operations of those rings, on top of which the function is bijective.

I think that an isomorphism can preserve any structure, so an isomorphism between groups preserves the group operation, isomorphisms between rings preserves ring operations, isomorphisms between vector spaces preserves scalar multiplication and vector addition, etc...

I could be wrong though.

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Hurkyl

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An isomorphism is defined to be a homomorphism with an inverse that is also a homomorphism.

A homomorphism of sets is just a function.

A homomorphism of topological spaces is just a continuous function.

A homomorphism of vector spaces is a linear transformation.

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matt grime

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jcsd

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matt grime said:

What's the difference? Are there bijective homomorphisms which are not invertible?

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No, but there are invertible homomorphisms which are not bijective.

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Hurkyl

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Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!

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Hurkyl said:Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!

Well, for a mapping to be an isomorphism, is it required that it's inverse be a homomorphism?

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No, but there are invertible homomorphisms which are not bijective.

I could've sworn that a function (homomorphism or not) was invertible iff it was bijective.

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Hurkyl

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Yes! In some cases, the inverse is automatically a homomorphism, but not in all. For example, consider the identity map [itex]\bar{\mathbb{R}} \rightarrow \mathbb{R}[/itex] where I am using [itex]\bar{\mathbb{R}}[/itex] to denote the discrete topology. This is clearly an invertible map, but it is not a homeomorphism. (Which is what we call isomorphisms when doing topology)Well, for a mapping to be an isomorphism, is it required that it's inverse be a homomorphism?

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jcsd

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Hurkyl said:Don't forget the bijective homomorphisms whose inverses aren't homomorphisms!

That does make more sense, looking up the catergory theory defintion of an isomorphism, a morphism isn't an isomorphism unless it is invertible and it's inverse is a morphism.

Still, the way that I informally think about homomorphisms, it is hard to imagine a bijective homomorphism whose inverse is not also a homomorphism. I'd guess that this is probably because in all the catergories which form my view of isomorphisms all bijective homomorphisms are isomorphisms.

Can you give me an example of a catergory with bijective homomorphisms which are not isomorphisms?

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matt grime

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Hurkyl just did: the identity mapping from any set with the discrete topology to itself with some other topology.

The map of differential manifolds from [0,1] to itself x-->2^2 is not invertible in the space of differential manifolds with diffeomorphisms (the inverse has no tangent at 0).

For Muzza: There are also plenty of isomorphisms in categories where it does not even make sense to start talking about bijections since the morphisms in no meaningful way act on elements of a set: morphisms are just arrows, they do not have to be maps on any underlying sets. This is one distinction between category theory and set theory.

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