Definition
Various inequivalent definitions of Kleene algebras and related structures have been given in the literature. See for a survey. Here we will give the definition that seems to be the most common nowadays.
A Kleene algebra is a set A together with two binary operations + : A × A → A and · : A × A → A and one function * : A → A, written as a + b, ab and a* respectively, so that the following axioms are satisfied.
- Associativity of + and ·: a + (b + c) = (a + b) + c and a(bc) = (ab)c for all a, b, c in A.
- Commutativity of +: a + b = b + a for all a, b in A
- Distributivity: a(b + c) = (ab) + (ac) and (b + c)a = (ba) + (ca) for all a, b, c in A
- Identity elements for + and ·: There exists an element 0 in A such that for all a in A: a + 0 = 0 + a = a. There exists an element 1 in A such that for all a in A: a1 = 1a = a.
- a0 = 0a = 0 for all a in A.
The above axioms define a semiring. We further require:
- + is idempotent: a + a = a for all a in A.
It is now possible to define a partial order ≤ on A by setting a ≤ b if and only if a + b = b (or equivalently: a ≤ b if and only if there exists an x in A such that a + x = b). With this order we can formulate the last two axioms about the operation *:
- 1 + a(a*) ≤ a* for all a in A.
- 1 + (a*)a ≤ a* for all a in A.
- if a and x are in A such that ax ≤ x, then a*x ≤ x
- if a and x are in A such that xa ≤ x, then x(a*) ≤ x
Intuitively, one should think of a + b as the "union" or the "least upper bound" of a and b and of ab as some multiplication which is monotonic, in the sense that a ≤ b implies ax ≤ bx. The idea behind the star operator is a* = 1 + a + aa + aaa + ... From the standpoint of programming language theory, one may also interpret + as "choice", · as "sequencing" and * as "iteration".
Read more about this topic: Kleene Algebra
Famous quotes containing the word definition:
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)