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

$\displaystyle \tilde{\psi}:G(N)\rightarrow G$