Kleene star

The Kleene star (or Kleene closure)an operation usedregular expressionsoperates either on sets strings or on setssymbols or characters. The application ofKleene star toset Vwritten as V*.

If V issetstrings then V*defined assmallest supersetV that contains ε (the empty string)is closed understring concatenation operation. This set can also be described assetstrings that can be made by concatenating zero or more strings from V.
If V issetsymbols or characters then V* issetall strings over symbolsV, includingempty string.

ExampleKleene star appliedsetstrings:

{"ab", "c"}* = {ε, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "ababc", "abcab", "abcc", "cabab", "cabc", "ccab", "ccc", ...}

ExampleKleene star appliedsetcharacters:

{'a', 'b', 'c'}* = {ε, "a", "b", "c", "aa", "ab", "ac", "ba", "bb", "bc", ...}

The Kleene staroften generalizedany monoid (M, .), that is,set Mbinary operation '.' on M such that

(closure)all abM, a . bM
(associativity)all a, bcM, (a . b) . c = a\ . (b . c)
(identity) therean eM such thatall a, a . e = e . a = a

If V issubsetM, then V*defined assmallest supersetV that contains ε (the empty string)is closed underoperation. V*then itselfmonoid,is calledmonoid generated by V. This isgeneralization ofKleene star discussed above sincesetall strings over some setsymbols formsmonoid (with string concatenation as binary operation).

The Kleene starnamed after Stephen Kleene (1909-1994) who introducedwhen describing certain automata (see regular expression).