If $\Sigma$ is an alphabet (a set of symbols), then the Kleene star of $\Sigma$, denoted $\Sigma^*$, is the set of all strings of finite length consisting of symbols in $\Sigma$, including the empty string $\lambda$.
If $S$ is a set of strings, then the Kleene star of $S$, denoted $S^*$, is the smallest superset of $S$ that contains $\lambda$ and is closed under the string concatenation operation. That is, $S^*$ is the set of all strings that can be generated by concatenating zero or more strings in $S$.
Question: why do we separate the definition into two distinct cases? How is one definition different than the other?