If is an alphabet (a set of symbols), then the Kleene star of , denoted , is the set of all strings of finite length consisting of symbols in , including the empty string . is also called the asterate.
If is a set of strings, then the Kleene star of , denoted , is the smallest superset of that contains and is closed under the string concatenation operation. That is, is the set of all strings that can be generated by concatenating zero or more strings in .
The definition of Kleene star can be generalized so that it operates on any monoid , where is a binary operation on the set . If is the identity element of and is a subset of , then is the smallest superset of that contains and is closed under .
, since there are no strings of finite length consisting of symbols in , so is the only element in .
If , then , since by definition, so .
If , then .
If , then
If , then
For any set , is the free monoid generated by .
Remark. There is an associated operation, called the Kleene plus, is defined for any set , such that is the smallest set containing such that is closed under the concatenation. In other words, .
|Date of creation||2013-03-22 12:26:58|
|Last modified on||2013-03-22 12:26:58|
|Last modified by||CWoo (3771)|