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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 created: h2013-03-21i by: hCWooi version: h39284i Privacy setting: h1i hDefinitioni h06F07i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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). 3