Kleene star
If $\mathrm{\Sigma}$ is an alphabet (a set of symbols), then the Kleene star of $\mathrm{\Sigma}$, denoted ${\mathrm{\Sigma}}^{*}$, is the set of all strings of finite length consisting of symbols in $\mathrm{\Sigma}$, 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

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

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

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

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

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If $S=\{ab,cd\}$, then ${S}^{*}=\{\lambda ,ab,cd,abab,abcd,cdab,cdcd,ababab,\mathrm{\dots}\}$
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  20130322 12:26:58 
Last modified on  20130322 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 