# Kleene star

Example of Kleene plus and Kleene star applied to the singleton set containing the empty string:

Strings form a monoid with concatenation as the binary operation and ε the identity element. The Kleene star is defined for any monoid, not just strings.
More precisely, let (*M*, ⋅) be a monoid, and *S* ⊆ *M*. Then *S*^{*} is the smallest submonoid of *M* containing *S*; that is, *S*^{*} contains the neutral element of *M*, the set *S*, and is such that if *x*,*y* ∈ *S*^{*}, then *x*⋅*y* ∈ *S*^{*}.

Furthermore, the Kleene star is generalized by including the *-operation (and the union) in the algebraic structure itself by the notion of complete star semiring.^{[4]}