- If
*V*is a set of strings then*V** is defined as the smallest superset of*V*that contains ε (the empty string) and is closed under the string concatenation operation. This set can also be described as the set of strings that can be made by concatenating zero or more strings from*V*. - 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.

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

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

- (closure) for all
*a*and*b*in*M*,*a*.*b*in*M* - (associativity) for all
*a*,*b*and*c*in*M*, (*a*.*b*) .*c*=*a*. (*b*.*c*) - (identity) there is an
*e*in*M*such that for all*a*,*a*.*e*=*e*.*a*=*a*

The Kleene star is named after Stephen Kleene (1909-1994) who introduced it when describing certain automata (see regular expression).

See also: