# Extending a homomorphism

#### Mauritzvdworm

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
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.

• Mauritzvdworm