Properties
Zero is the smallest element: 0 ≤ a for all a in A.
The sum a + b is the least upper bound of a and b: we have a ≤ a + b and b ≤ a + b and if x is an element of A with a ≤ x and b ≤ x, then a + b ≤ x. Similarly, a1 + ... + an is the least upper bound of the elements a1, ..., an.
Multiplication and addition are monotonic: if a ≤ b, then a + x ≤ b + x, ax ≤ bx and xa ≤ xb for all x in A.
Regarding the * operation, we have 0* = 1 and 1* = 1, that * is monotonic (a ≤ b implies a* ≤ b*), and that an ≤ a* for every natural number n. Furthermore, (a*)(a*) = a*, (a*)* = a*, and a ≤ b* if and only if a* ≤ b*.
If A is a Kleene algebra and n is a natural number, then one can consider the set Mn(A) consisting of all n-by-n matrices with entries in A. Using the ordinary notions of matrix addition and multiplication, one can define a unique *-operation so that Mn(A) becomes a Kleene algebra.
Read more about this topic: Kleene Algebra
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)