Bijective Homomorphisms and Isomorphisms

In summary, universal algebra and algebraic category both exist, but the definition of algebra in each case is different.
  • #1
WWGD
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Hi All,

Let A,B be algebraic structures and let h A-->B be a bijective homomorphism.
Is h an isomorphism? In topology, we have continuous bijections that are not homeomorphisms,
(similar in Functional Analysis )so I wondered if the "same" was possible in Algebra. I assume if there is a counterexample, it requires an infinite set in the construction, or some result in order theory, or some issue with torsion .

Thanks,

WWGD: "What Would Gauss Do?".
 
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  • #2
Depends on what you mean with "algebra".

There are two ways I could generalize algebra. The first way is through universal algebra. That generalizes a lot of algebraic objects such as modules, groups, rings, etc. A free course can be found here: http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html A isomorphism there is defined as a bijective structure-preserving map. One can indeed prove that for all universal algebras, all such isomorphisms are isomorphisms in the categorical setting.

Another definition of algebra would consist of the notion of algebraic category. For example, see the "joy of cats": http://katmat.math.uni-bremen.de/acc/acc.pdf chapter VI
In particular, see proposition 23.7. That implies that all bimorphisms (= both mono and epimorphisms) are isomorphisms. In particular, for all usual algebraic categories (usual = concrete category over set), the surjective and injective morphisms will indeed be isomorphisms.

It is interesting to note that the category of all compact Hausdorff spaces is also considered algebraic. That the bijective continuous functions between compact Hausdorff spaces are homeomorphisms is well known. But the algebraicity of the compact Hausdorff spaces predicts somehow that the compact Hausdorff spaces should be determined by some "conventional" algebraic structure. This algebraic structure is the ring of continuous functions: http://en.wikipedia.org/wiki/Continuous_functions_on_a_compact_Hausdorff_space
This excellent book gives more information: https://www.amazon.com/dp/1258632012/?tag=pfamazon01-20
 
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  • #3
O.K, thanks, micromass.
 

1. What is a bijective homomorphism?

A bijective homomorphism is a type of mathematical function that preserves the structure and operations of a group or algebraic structure. It is both injective (one-to-one) and surjective (onto), meaning that each element in the codomain has a unique preimage in the domain. In other words, a bijective homomorphism is a one-to-one correspondence between two structures that preserves their algebraic properties.

2. How is a bijective homomorphism different from a regular homomorphism?

A regular homomorphism only needs to preserve the structure and operations of a group or algebraic structure, without requiring it to be one-to-one or onto. This means that a regular homomorphism can map multiple elements to the same element in the codomain, while a bijective homomorphism must have a unique preimage for each element in the codomain.

3. What is the significance of bijective homomorphisms?

Bijective homomorphisms are important in mathematics because they allow us to compare and understand different algebraic structures by showing that they are essentially the same. By preserving the structure and operations of a group or algebraic structure, bijective homomorphisms help us identify patterns and relationships between seemingly different mathematical objects.

4. What is an isomorphism?

An isomorphism is a bijective homomorphism between two algebraic structures that preserves not only their structure and operations, but also their identities. In other words, an isomorphism is a bijective homomorphism that also respects the special properties of the structures, such as the existence of an identity element or the commutativity of operations.

5. How do you prove that two structures are isomorphic?

To prove that two structures are isomorphic, you must show that there exists a bijective homomorphism between them. This can be done by constructing a mapping between the two structures and demonstrating that it is both one-to-one and onto, as well as preserving the operations and identities of the structures. Alternatively, you can also show that the two structures are equivalent by finding a way to transform one into the other while preserving their algebraic properties.

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