for all .
For a given alphabet , the set of all languages over , as well as the set of all regular languages over , are examples of Kleene algebras. Similarly, sets of regular expressions (regular sets) over are a form (or close variant) of a Kleene algebra: let be the set of all regular sets over a set of alphabets. Then is a Kleene algebra if we identify as , the singleton containing the empty string as , concatenation operation as , the union operation as , and the Kleene star operation as . For example, let be a set of regular expression, then
Adding on both sides and we have
The other conditions are checked similarly.
Remark. There is another notion of a Kleene algebra, which arises from lattices. For more detail, see here (http://planetmath.org/KleeneAlgebra2).
|Date of creation||2013-03-22 12:27:51|
|Last modified on||2013-03-22 12:27:51|
|Last modified by||CWoo (3771)|