Kleene Algebra - Definition

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 × AA and · : A × AA and one function * : AA, 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 ab if and only if a + b = b (or equivalently: ab 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*)aa* for all a in A.
  • if a and x are in A such that axx, then a*xx
  • if a and x are in A such that xax, 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 ab implies axbx. 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".

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