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Thread: Extending a homomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    Extending a homomorphism

    Suppose $\displaystyle N$ is a cancellative abelian semigroup with zero element and we have the homomorphism
    $\displaystyle \psi:N\rightarrow G$
    where $\displaystyle G$ is a abelian group, how do I extend this homomorphism to the whole of $\displaystyle G(N)$ where $\displaystyle G(N)$ is the envelopping Gothendieck group of $\displaystyle N$ to obtian the homomorphism
    $\displaystyle \tilde{\psi}:G(N)\rightarrow G$
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  2. #2
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    Quote Originally Posted by Mauritzvdworm View Post
    Suppose $\displaystyle N$ is a cancellative abelian semigroup with zero element and we have the homomorphism
    $\displaystyle \psi:N\rightarrow G$
    where $\displaystyle G$ is a abelian group, how do I extend this homomorphism to the whole of $\displaystyle G(N)$ where $\displaystyle G(N)$ is the enveloping Gothendieck group of $\displaystyle N$ to obtain the homomorphism
    $\displaystyle \tilde{\psi}:G(N)\rightarrow G$
    Elements of $\displaystyle G(N)$ are (equivalence classes of) differences of pairs of elements of N. If $\displaystyle m-n\in G(N)$ then define $\displaystyle \tilde{\psi}(m-n) = \psi(m) - \psi(n)$. You need to show that this is well-defined. In other words, you must show that if $\displaystyle m_1-n_1 = m_2-n_2$ then $\displaystyle \psi(m_1)-\psi(n_1) = \psi(m_2)-\psi(n_2)$. Once you have done this, it is more or less obvious that $\displaystyle \tilde{\psi}$ is a homomorphism.
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