Does there exist a commutative semigroup $S$ with the following (additively denoted) properties?
- For all $x \in S$, the function $S \rightarrow S$ given by $y \mapsto x+y$ is a bijection.
- $S$ has no identity element; that is, there is no $0 \in S$ such that for all $x \in S$, $x+0=x.$
- $S$ is non-empty. (This is to rule out the trivial example of $S=\emptyset$, thanks to Najib Idrissi for pointing this out.)
Of course, if 2 is replaced by its negation (namely, that there is an identity element), then examples are easy to find; in fact, these are precisely the Abelian groups.