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quantale∗
CWoo†
2013-03-21 22:41:22
A quantale Q is a set with three binary operations on it: ∧, ∨, and ·, such
that
1. (Q, ∧, ∨) is a complete lattice (with 0 as the bottom and 1 as the top),
and
2. (Q, ·) is a monoid (with 10 as the identity with respect to ·), such that
3. · distributes over arbitrary joins; that is, for any a ∈ Q and any subset
S ⊆ Q,
_ _
_ _
a·
S = {a · s | s ∈ S} and
S · a = {s · a | s ∈ S}.
It is sometimes convenient to drop the multiplication symbol, when there is no
confusion. So instead of writing a · b, we write ab.
The most obvious example of a quantale comes from ring theory. Let R be
a commutative ring with 1. Then L(R), the lattice of ideals of R, is a quantale.
Proof. In addition to being a (complete) lattice, L(R) has an inherent multiplication operation induced by the multiplication on R, namely,
n
X
ri si | ri ∈ I and si ∈ J, n ∈ N},
IJ := {
i=1
making it into a semigroup under the multiplication.
W
Now, let S = {Ii | i ∈ N } be a set of ideals
W of R and let I = S. If J
is any ideal of R, we want to show that IJ = {Ii J |Wi ∈ N } and, since R is
commutative, we would havePthe other equality JI = {JIi | i ∈ N }. To see
this, let a ∈ IJ. Then aS=
ri si with ri ∈ I and si ∈ J. Since
S each ri is a
finite sum
of
elements
of
S,
r
s
is
a
finite
sum
of
elements
of
{Ii J | i ∈ N },
i
i
W
W
so a W
∈ {Ii J | i ∈ N }. This shows IJ ⊆ {Ii J | i ∈ N }. Conversely, if
a ∈ {Ii J | i ∈ N }, then a can be written as a finite sum of elements of
∗ hQuantalei
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S
{Ii J | i ∈ N }. In turn, each of these additive components is a finite sum of
products of the form rk sk , where rk ∈ Ii for some i, and sk ∈ J. As a result,
a is a finite
W sum of elements of the form rk sk , so a ∈ IJ and we have the other
inclusion {Ii J | i ∈ N } ⊆ IJ.
Finally, we observe that R is the multiplicative identity in L(R), as IR =
RI = I for all I ∈ L(R). This completes the proof.
Remark. In the above example, notice that IJ ≤ I and IJ ≤ J, and we
actually have IJ ≤ I ∧ J. In particular, I 2 ≤ I. With an added condition, this
fact can be characterized in an arbitrary quantale (see below).
Properties. Let Q be a quantale.
1. Multiplication is monotone in each argument. This means that if a, b ∈ Q,
then a ≤ b implies that ac ≤ bc and ca ≤ cb for all c ∈ Q. This is easily
verified. For example, if a ≤ b, then ac ∨ bc = (a ∨ b)c = bc, so ac ≤ bc.
So a quantale is a partially ordered semigroup, and in fact, an l-monoid
(an l-semigroup and a monoid at the same time).
2. If 1 = 10 , then ab ≤ a ∧ b: since a ≤ 1, then ab ≤ a1 = a10 = a;
similarly, b ≤ ab. In particular, the bottom 0 is also the multiplicative
zero: a0 ≤ a ∧ 0 = 0, and 0a = 0 similarly.
3. Actually, a0 = 0aW= 0 is true even without
1W
= 10 : since
W
W a∅ = {ab | b ∈
∅} = ∅ and 0 := ∅, we have a0 = a ∅ = a∅ = ∅ = 0. Similarly
0a = 0. So a quantale is a semiring, if ∨ is identified as + (with 0 as the
additive identity), and · is again · (with 10 the multiplicative identity).
4. Viewing quantale Q now as a semiring, we see in fact that Q is an idempotent semiring, since a + a = a ∨ a = a.
5. Now, view QWas an i-semiring. For each a ∈ Q, let S = {10 , a, a2 , . . .} and
define a∗ = S. We observe some basic properties
W
W
• 1W0 + aa∗ = a∗ : since
10 ∨ (a S) = 10 ∨ ( {a10 , aa, aa2 , . . .}) =
W
{10 , a, a2 , . . .} = S = a∗
• 10 + a∗ a = a∗ as well
• if ab ≤ b, then a∗ b ≤ W
b: by induction on n, we have an b ≤ b whenever
∗
a ≤ b, so that a b = {an b | n ∈ N ∪ {0}} ≤ b.
• similarly, if ba ≤ b, then ba∗ ≤ b
All of the above properties satisfy the conditions for an i-semiring to be a
Kleene algebra. For this reason, a quantale is sometimes called a standard
Kleene algebra.
6. Call the multiplication idempotent if each element is an idempotent with
respect to the multiplication: aa = a for any a ∈ Q. If · is idempotent
and 1 = 10 , then · = ∧. In other words, ab = a ∧ b.
2
Proof. As we have seen, ab ≤ a ∧ b in the 2 above. Now, suppose c ≤ a ∧ b.
Then c ≤ a and c ≤ b, so c = c2 ≤ cb ≤ ab. So ab is the greatest lower
bound of a and b, i.e., ab = a ∧ b. This also means that ba = b ∧ a =
a ∧ b = ab.
7. In fact, a locale is a quantale if we define · := ∧. Conversely, a quantale
where · is idempotent and 1 = 10 is a locale.
Proof. If Q is a locale with · = ∧, then aa = a ∧ a = a and a1 = a ∧ 1 =
a = 1 ∧ a = 1a, implying 1 = 10 . The infinite distributivity of · over ∨
is just a restatement of the infinite distributivity of ∧ over ∨ in a locale.
0
Conversely,
W if · is idempotent
W
W and 1 = 1 , then
W · = ∧ as shown previously,
so
a
∧
(
S)
=
a(
S)
=
{as
|
s
∈
S}
=
{a ∧ s | s ∈ S}. Similarly
W
W
( S) ∧ a = {s ∧ a | s ∈ S}. Therefore, Q is a locale.
Remark. A quantale homomorphism between two quantales is a complete
lattice homomorphism and a monoid homomorphism at the same time.
References
[1] S. Vickers, Topology via Logic, Cambridge University Press, Cambridge
(1989).
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