Bijective Homomorphisms and Isomorphisms

  • Thread starter WWGD
  • Start date

WWGD

Science Advisor
Gold Member
4,246
1,805
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?".
 
21,993
3,266
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
 
Last edited by a moderator:

WWGD

Science Advisor
Gold Member
4,246
1,805
O.K, thanks, micromass.
 

Related Threads for: Bijective Homomorphisms and Isomorphisms

  • Posted
Replies
11
Views
12K
Replies
7
Views
3K
  • Posted
Replies
1
Views
2K
Replies
7
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top