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**Drexel28** Additive in terms of abelian? A lot of this is actually unecessary. For example if you have a semigroup $\displaystyle (S,\ast)$ (i.e. just a set with an associative binary operation $\displaystyle \ast:S\times S\to S$) then any identity would have to be unique, for if $\displaystyle i,i'$ were both identities then $\displaystyle i=i\ast i'$ because $\displaystyle i'$ is an identity and $\displaystyle i\ast i'=i'$ since $\displaystyle i$ is an identity. Make sense? Now use this fact to prove that $\displaystyle -x$ is unique, i.e. assume that $\displaystyle y,z$ are both inverses of $\displaystyle x$ and note that $\displaystyle x+y$ and $\displaystyle x+z$ are identities and so $\displaystyle x+y=x+z$...so