Extending a homomorphism

Aug 2009
122
19
Pretoria
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
\(\displaystyle \tilde{\psi}:G(N)\rightarrow G\)
 

Opalg

MHF Hall of Honor
Aug 2007
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Leeds, UK
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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