Kleene star
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 andis closed under
the string concatenation operation. That is, is theset 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 anymonoid , 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 thatcontains and is closed under .
Examples
- •
, since there are no strings of finite length consisting of symbols in , so is the only element in .
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If , then , since by definition, so .
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If , then .
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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, .
Title | Kleene star |
Canonical name | KleeneStar |
Date of creation | 2013-03-22 12:26:58 |
Last modified on | 2013-03-22 12:26:58 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 9 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 20M35 |
Classification | msc 68Q70 |
Synonym | asterate |
Related topic | Alphabet |
Related topic | String |
Related topic | RegularExpression |
Related topic | KleeneAlgebra |
Related topic | Language |
Related topic | Convolution2 |
Related topic | WeightStrings |
Related topic | WeightEnumerator |
Defines | Kleene plus |