is a complete lattice (with as the bottom and as the top), and
is a monoid (with as the identity with respect to ), such that
distributes over arbitrary joins; that is, for any and any subset ,
It is sometimes convenient to drop the multiplication symbol, when there is no confusion. So instead of writing , we write .
making it into a semigroup under the multiplication.
Now, let be a set of ideals of and let . If is any ideal of , we want to show that and, since is commutative, we would have the other equality . To see this, let . Then with and . Since each is a finite sum of elements of , is a finite sum of elements of , so . This shows . Conversely, if , then can be written as a finite sum of elements of . In turn, each of these additive components is a finite sum of products of the form , where for some , and . As a result, is a finite sum of elements of the form , so and we have the other inclusion .
Finally, we observe that is the multiplicative identity in , as for all . This completes the proof. ∎
Remark. In the above example, notice that and , and we actually have . In particular, . With an added condition, this fact can be characterized in an arbitrary quantale (see below).
Properties. Let be a quantale.
If , then : since , then ; similarly, . In particular, the bottom is also the multiplicative zero: , and similarly.
Viewing quantale now as a semiring, we see in fact that is an idempotent semiring, since .
Now, view as an i-semiring. For each , let and define . We observe some basic properties
if , then : by induction on , we have whenever , so that .
similarly, if , then
As we have seen, in the 2 above. Now, suppose . Then and , so . So is the greatest lower bound of and , i.e., . This also means that . ∎
In fact, a locale is a quantale if we define . Conversely, a quantale where is idempotent and is a locale.
If is a locale with , then and , implying . The infinite distributivity of over is just a restatement of the infinite distributivity of over in a locale. Conversely, if is idempotent and , then as shown previously, so . Similarly . Therefore, is a locale. ∎
- 1 S. Vickers, Topology via Logic, Cambridge University Press, Cambridge (1989).
|Date of creation||2013-03-22 17:00:08|
|Last modified on||2013-03-22 17:00:08|
|Last modified by||CWoo (3771)|
|Synonym||standard Kleene algebra|