Results 1 to 2 of 2

Math Help - Extending a homomorphism

  1. #1
    Member Mauritzvdworm's Avatar
    Joined
    Aug 2009
    From
    Pretoria
    Posts
    122

    Extending a homomorphism

    Suppose N is a cancellative abelian semigroup with zero element and we have the homomorphism
    \psi:N\rightarrow G
    where G is a abelian group, how do I extend this homomorphism to the whole of G(N) where G(N) is the envelopping Gothendieck group of N to obtian the homomorphism
    \tilde{\psi}:G(N)\rightarrow G
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Mauritzvdworm View Post
    Suppose N is a cancellative abelian semigroup with zero element and we have the homomorphism
    \psi:N\rightarrow G
    where G is a abelian group, how do I extend this homomorphism to the whole of G(N) where G(N) is the enveloping Gothendieck group of N to obtain the homomorphism
    \tilde{\psi}:G(N)\rightarrow G
    Elements of G(N) are (equivalence classes of) differences of pairs of elements of N. If m-n\in G(N) then define \tilde{\psi}(m-n) = \psi(m) - \psi(n). You need to show that this is well-defined. In other words, you must show that if m_1-n_1 = m_2-n_2 then \psi(m_1)-\psi(n_1) = \psi(m_2)-\psi(n_2). Once you have done this, it is more or less obvious that \tilde{\psi} is a homomorphism.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 8
    Last Post: June 26th 2011, 04:00 AM
  2. [SOLVED] Extending an exp function to be odd.
    Posted in the Calculus Forum
    Replies: 7
    Last Post: May 29th 2011, 12:50 PM
  3. Extending a linearly independent set
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: February 11th 2011, 05:24 AM
  4. Extending to closed space
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 29th 2009, 04:10 PM
  5. [SOLVED] Extending Differentiation: Chain Rule
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 25th 2008, 03:13 AM

Search Tags


/mathhelpforum @mathhelpforum