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Kleene star
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 $\lambda$. ${}^{*}$ is also called the asterate.
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 $S$.
The definition of Kleene star can be generalized so that it operates on any monoid $(M,++)$, where $++$ is a binary operation on the set $M$. If $e$ is the identity element of $(M,++)$ and $S$ is a subset of $M$, then $S^{*}$ is the smallest superset of $S$ that contains $e$ and is closed under $++$.
Examples

$\varnothing^{*}=\{\lambda\}$, since there are no strings of finite length consisting of symbols in $\varnothing$, so $\lambda$ is the only element in $\varnothing^{*}$.

If $E=\{\lambda\}$, then $E^{*}=E$, since $\lambda a=a\lambda=a$ by definition, so $\lambda\lambda=\lambda$.

If $A=\{a\}$, then $A^{*}=\{\lambda,a,aa,aaa,\ldots\}$.

If $\Sigma=\left\{a,b\right\}$, then $\Sigma^{*}=\left\{\lambda,a,b,aa,ab,ba,bb,aaa,\dots\right\}$

If $S=\left\{ab,cd\right\}$, then $S^{*}=\left\{\lambda,ab,cd,abab,abcd,cdab,cdcd,ababab,\dots\right\}$
For any set $S$, $S^{*}$ is the free monoid generated by $S$.
Remark. There is an associated operation, called the Kleene plus, is defined for any set $S$, such that $S^{+}$ is the smallest set containing $S$ such that $S^{+}$ is closed under the concatenation. In other words, $S^{+}=S^{*}\{\lambda\}$.
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