# Bijective Homomorphisms and Isomorphisms

#### WWGD

Science Advisor
Gold Member
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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#### micromass

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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#### WWGD

Science Advisor
Gold Member
O.K, thanks, micromass.

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