A quantale Q is a set with three binary operationsMathworldPlanetmath on it: ,, and , such that

  1. 1.

    (Q,,) is a complete latticeMathworldPlanetmath (with 0 as the bottom and 1 as the top), and

  2. 2.

    (Q,) is a monoid (with 1 as the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath with respect to ), such that

  3. 3.

    distributes over arbitrary joins; that is, for any aQ and any subset SQ,

    a(S)={assS} and (S)a={sasS}.

It is sometimes convenient to drop the multiplication symbol, when there is no confusion. So instead of writing ab, 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.


In addition to being a (completePlanetmathPlanetmathPlanetmathPlanetmath) latticeMathworldPlanetmath, L(R) has an inherent multiplication operationMathworldPlanetmath induced by the multiplication on R, namely,

IJ:={i=1nrisiriI and siJn},

making it into a semigroupPlanetmathPlanetmath under the multiplication.

Now, let S={IiiN} be a set of ideals of R and let I=S. If J is any ideal of R, we want to show that IJ={IiJiN} and, since R is commutativePlanetmathPlanetmathPlanetmath, we would have the other equality JI={JIiiN}. To see this, let aIJ. Then a=risi with riI and siJ. Since each ri is a finite sum of elements of S, risi is a finite sum of elements of {IiJiN}, so a{IiJiN}. This shows IJ{IiJiN}. Conversely, if a{IiJiN}, then a can be written as a finite sum of elements of {IiJiN}. In turn, each of these additive components is a finite sum of productsMathworldPlanetmath of the form rksk, where rkIi for some i, and skJ. As a result, a is a finite sum of elements of the form rksk, so aIJ and we have the other inclusion {IiJiN}IJ.

Finally, we observe that R is the multiplicative identityPlanetmathPlanetmath in L(R), as IR=RI=I for all IL(R). This completes the proof. ∎

Remark. In the above example, notice that IJI and IJJ, and we actually have IJIJ. In particular, I2I. With an added condition, this fact can be characterized in an arbitrary quantale (see below).

Properties. Let Q be a quantale.

  1. 1.

    Multiplication is monotone in each argument. This means that if a,bQ, then ab implies that acbc and cacb for all cQ. This is easily verified. For example, if ab, then acbc=(ab)c=bc, so acbc. 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. 2.

    If 1=1, then abab: since a1, then aba1=a1=a; similarly, bab. In particular, the bottom 0 is also the multiplicative zero: a0a0=0, and 0a=0 similarly.

  3. 3.

    Actually, a0=0a=0 is true even without 1=1: since a={abb}= and 0:=, we have a0=a=a==0. Similarly 0a=0. So a quantale is a semiringMathworldPlanetmath, if is identified as + (with 0 as the additive identity), and is again (with 1 the multiplicative identity).

  4. 4.

    Viewing quantale Q now as a semiring, we see in fact that Q is an idempotent semiring, since a+a=aa=a.

  5. 5.

    Now, view Q as an i-semiring. For each aQ, let S={1,a,a2,} and define a*=S. We observe some basic properties

    • 1+aa*=a*: since 1(aS)=1({a1,aa,aa2,})={1,a,a2,}=S=a*

    • 1+a*a=a* as well

    • if abb, then a*bb: by inductionMathworldPlanetmath on n, we have anbb whenever ab, so that a*b={anbn{0}}b.

    • similarly, if bab, 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. 6.

    Call the multiplication idempotentMathworldPlanetmathPlanetmath if each element is an idempotent with respect to the multiplication: aa=a for any aQ. If is idempotent and 1=1, then =. In other words, ab=ab.


    As we have seen, abab in the 2 above. Now, suppose cab. Then ca and cb, so c=c2cbab. So ab is the greatest lower boundMathworldPlanetmath of a and b, i.e., ab=ab. This also means that ba=ba=ab=ab. ∎

  7. 7.

    In fact, a locale is a quantale if we define :=. Conversely, a quantale where is idempotent and 1=1 is a locale.


    If Q is a locale with =, then aa=aa=a and a1=a1=a=1a=1a, implying 1=1. The infiniteMathworldPlanetmath distributivity of over is just a restatement of the infinite distributivity of over in a locale. Conversely, if is idempotent and 1=1, then = as shown previously, so a(S)=a(S)={assS}={assS}. Similarly (S)a={sasS}. 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.


  • 1 S. Vickers, Topology via Logic, Cambridge University Press, Cambridge (1989).
Title quantale
Canonical name Quantale
Date of creation 2013-03-22 17:00:08
Last modified on 2013-03-22 17:00:08
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 06F07
Synonym standard Kleene algebra
Defines quantale homomorphism