Kleene star

If Σ is an alphabet (a set of symbols), then the Kleene star of Σ, denoted Σ*, is the set of all strings of finite length consisting of symbols in Σ, including the empty string λ. * is also called the asterate.

If S is a set of strings, then the Kleene star of S, denoted S*, is the smallest supersetMathworldPlanetmath of S that contains λ and is closed underPlanetmathPlanetmath 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 operationMathworldPlanetmath on the set M. If e is the identity elementMathworldPlanetmath of (M,++) and S is a subset of M, then S* is the smallest superset of S that contains e and is closed under ++.


  • *={λ}, since there are no strings of finite length consisting of symbols in , so λ is the only element in *.

  • If E={λ}, then E*=E, since λa=aλ=a by definition, so λλ=λ.

  • If A={a}, then A*={λ,a,aa,aaa,}.

  • If Σ={a,b}, then Σ*={λ,a,b,aa,ab,ba,bb,aaa,}

  • If S={ab,cd}, then S*={λ,ab,cd,abab,abcd,cdab,cdcd,ababab,}

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 concatenationMathworldPlanetmath. In other words, S+=S*-{λ}.

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 LanguagePlanetmathPlanetmath
Related topic Convolution2
Related topic WeightStrings
Related topic WeightEnumerator
Defines Kleene plus