Kleene star

Remarks

Example: for $L = \set{ab}$, $L^* = \set{\varepsilon, ab, abab, ababab, \dots}$. The string $abab$ is in $L^2$, with $w_1 = w_2 = ab$, so it is in $L^*$. Non-example: $aba \notin L^*$, since every member of $L^*$ has even length. Edge cases: $\emptyset^* = \set{\varepsilon}$, since $\emptyset^0 = \set{\varepsilon}$ and $\emptyset^n = \emptyset$ for $n \ge 1$. The Kleene plus is $L^+ = \bigcup_{n \ge 1} L^n$, and $L^* = L^+ \cup \set{\varepsilon}$. For an alphabet $\Sigma$, viewed as the language of strings of length $1$, $\Sigma^*$ is the set of all strings over $\Sigma$. $L^*$ is the smallest language that contains $\varepsilon$ and $L$ and is closed under concatenation.