Kleene star

Unary operation on sets of strings, used in regular expressions for "zero or more repetitions"

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 SM. 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,yS*, then xyS*.

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]