Kleene star


In mathematical logic and computer science, the Kleene star is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics
it is more commonly known as the free monoid construction. The application of the Kleene star to a set V is written as V*. It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more".
  1. If V is a set of strings, then V* is defined as the smallest superset of V that contains the empty string ε and is closed under the string concatenation operation.
  2. If V is a set of symbols or characters, then V* is the set of all strings over symbols in V, including the empty string ε.
The set V* can also be described as the set of finite-length strings that can be generated by concatenating arbitrary elements of V, allowing the use of the same element multiple times. If V is either the empty set ∅ or the singleton set, then V* = ; if V is any other finite set or countably infinite set, then V* is a countably infinite set.
The operators are used in rewrite rules for generative grammars.

Definition and notation

Given a set V
define
and define recursively the set
If V is a formal language, then Vi, the i-th power of the set V, is a shorthand for the concatenation of set V with itself i times. That is, Vi can be understood to be the set of all strings that can be represented as the concatenation of i strings in V.
The definition of Kleene star on V is
Notice that the Kleene star operator is an idempotent unary operator: * = V* for any set V of strings or characters.

Kleene plus

In some formal language studies, a variation on the Kleene star operation called the Kleene plus is used. The Kleene plus omits the V0 term in the above union. In other words, the Kleene plus on V is
For every set L, the Kleene plus of L equals the concatenation of L with L*; this holds because every element of L+ must either be composed from one element of L and finitely many non-empty terms in L or is just an element of L.
Conversely, L* = ∪ L+.

Examples

Example of Kleene star applied to set of strings:
Example of Kleene plus applied to set of characters:
Kleene star applied to the same character set:
Example of Kleene star applied to the empty set:
Example of Kleene plus applied to the empty set:
where concatenation is an associative and noncommutative product, sharing these properties with the Cartesian product of sets.
Example of Kleene plus and Kleene star applied to the singleton set containing the empty string:

Generalization

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 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 in the algebraic structure itself by the notion of complete star semiring.