Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bijective Homomorphisms and Isomorphisms

  1. Jul 17, 2014 #1

    WWGD

    User Avatar
    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?".
     
  2. jcsd
  3. Jul 17, 2014 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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/Rings-Continuous-Functions-University-Mathematics/dp/1258632012/
     
    Last edited by a moderator: May 6, 2017
  4. Jul 18, 2014 #3

    WWGD

    User Avatar
    Science Advisor
    Gold Member

    O.K, thanks, micromass.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook