# 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 $(M,+\hskip-4.3pt+)$, where $+\hskip-4.3pt+$ is a binary operation  on the set $M$. If $e$ is the identity element  of $(M,+\hskip-4.3pt+)$ and $S$ is a subset of $M$, then $S^{*}$ is the smallest superset of $S$ that contains $e$ and is closed under $+\hskip-4.3pt+$.

## 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, $S^{+}=S^{*}-\{\lambda\}$.

 Title Kleene star Canonical name KleeneStar Date of creation 2013-03-22 12:26:58 Last modified on 2013-03-22 12:26:58 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 9 Author CWoo (3771) Entry type Definition Classification msc 20M35 Classification msc 68Q70 Synonym asterate Related topic Alphabet Related topic String Related topic RegularExpression Related topic KleeneAlgebra Related topic Language  Related topic Convolution2 Related topic WeightStrings Related topic WeightEnumerator Defines Kleene plus