Let $(S,+,\cdot)$ be a semiring with or without 0 but necessarily with 1. Let $f: S \rightarrow S$ be defined by $f(k)=k+k$. What is the weakest possible extra assumption I need to make on $S$ so that $f$ is injective.
"Weak" will mean an assumption that achieves injectivity but does not imply the following sufficient condition, multiplicative cancellation: $\forall a,b,c \in S$, $a\cdot b=a\cdot c \Rightarrow b=c$. This is sufficient since
$$f(k)=f(k')$$ $$\Rightarrow k+k=k'+k'$$ $$\Rightarrow (1+1)k=(1+1)k'$$ $$\Rightarrow k=k'$$