Semigroups -- Underlying Set Non Empty or it can be Empty?

Question

Many authors consider Semigroups being Non Empty sets. Others include empty sets as Semigroups. What is the rationale behind both choices ?

Answer 1

This is largely a matter of convenience. I can think of only one reason for letting a semigroup be empty: if $f:G\to H$ is a morhpism of semigroups and $H$ happens to be a semigroup with unit $e$, then $\operatorname{ker}f=\{x\in H | f(x)=e\}$ is naturally a semigroup --- unless it is empty (which may happen if $G$ does not have a unit). To include this case in the general framework, one might allow a semigroup to be empty.