quantale

quantale

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

1

′

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

a

⁢

b

.

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,

I

⁢

J

:=

{

∑

i

=

1

n

r

i

⁢

s

i

∣

r

i

∈

I

⁢

and

⁢

s

i

∈

J

⁢

,

⁢

n

∈

ℕ

}

,

making it into a

semigroup

under the multiplication.

Now, let

S

=

{

I

i

∣

i

∈

N

}

be a set of ideals of

R

and let

I

=

⋁

S

. If

J

is any ideal of

R

, we want to show that

I

⁢

J

=

⋁

{

I

i

⁢

J

∣

i

∈

N

}

and, since

R

is

commutative

, we would have the other equality

J

⁢

I

=

⋁

{

J

⁢

I

i

∣

i

∈

N

}

. To see this, let

a

∈

I

⁢

J

. Then

a

=

∑

r

i

⁢

s

i

with

r

i

∈

I

and

s

i

∈

J

. Since each

r

i

is a finite sum of elements of

⋃

S

,

r

i

⁢

s

i

is a finite sum of elements of

⋃

{

I

i

⁢

J

∣

i

∈

N

}

, so

a

∈

⋁

{

I

i

⁢

J

∣

i

∈

N

}

. This shows

I

⁢

J

⊆

⋁

{

I

i

⁢

J

∣

i

∈

N

}

.

Conversely

, if

a

∈

⋁

{

I

i

⁢

J

∣

i

∈

N

}

, then

a

can be written as a finite sum of elements of

⋃

{

I

i

⁢

J

∣

i

∈

N

}

. In turn, each of these

additive

components

is a finite sum of

products

of the form

r

k

⁢

s

k

, where

r

k

∈

I

i

for some

i

, and

s

k

∈

J

. As a result,

a

is a finite sum of elements of the form

r

k

⁢

s

k

, so

a

∈

I

⁢

J

and we have the other

inclusion

⋁

{

I

i

⁢

J

∣

i

∈

N

}

⊆

I

⁢

J

.

Finally, we observe that

R

is the

multiplicative identity

in

L

⁢

(

R

)

, as

I

⁢

R

=

R

⁢

I

=

I

for all

I

∈

L

⁢

(

R

)

. This completes the proof.
∎

Remark

. In the above example, notice that

I

⁢

J

≤

I

and

I

⁢

J

≤

J

, and we actually have

I

⁢

J

≤

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

a

⁢

c

≤

b

⁢

c

and

c

⁢

a

≤

c

⁢

b

for all

c

∈

Q

. This is easily verified. For example, if

a

≤

b

, then

a

⁢

c

∨

b

⁢

c

=

(

a

∨

b

)

⁢

c

=

b

⁢

c

, so

a

⁢

c

≤

b

⁢

c

. 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

=

1

′

, then

a

⁢

b

≤

a

∧

b

: since

a

≤

1

, then

a

⁢

b

≤

a

⁢

1

=

a

⁢

1

′

=

a

; similarly,

b

≤

a

⁢

b

. In particular, the bottom

0

is also the multiplicative zero:

a

⁢

0

≤

a

∧

0

=

0

, and

0

⁢

a

=

0

similarly.

3.

Actually,

a

⁢

0

=

0

⁢

a

=

0

is true even without

1

=

1

′

: since

a

⁢

∅

=

{

a

⁢

b

∣

b

∈

∅

}

=

∅

and

0

:=

⋁

∅

, we have

a

⁢

0

=

a

⁢

⋁

∅

=

⋁

a

⁢

∅

=

⋁

∅

=

0

. Similarly

0

⁢

a

=

0

. So a quantale is a

semiring

, if

∨

is identified as

+

(with

0

as the

additive identity

), and

⋅

is again

⋅

(with

1

′

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

Q

as an i-semiring. For each

a

∈

Q

, let

S

=

{

1

′

,

a

,

a

2

,

…

}

and define

a

*

=

⋁

S

. We observe some basic properties

–

1

′

+

a

⁢

a

*

=

a

*

: since

1

′

∨

(

a

⁢

⋁

S

)

=

1

′

∨

(

⋁

{

a

⁢

1

′

,

a

⁢

a

,

a

⁢

a

2

,

…

}

)

=

⋁

{

1

′

,

a

,

a

2

,

…

}

=

⋁

S

=

a

*

–

1

′

+

a

*

⁢

a

=

a

*

as well

–

if

a

⁢

b

≤

b

, then

a

*

⁢

b

≤

b

: by

induction

on

n

, we have

a

n

⁢

b

≤

b

whenever

a

≤

b

, so that

a

*

⁢

b

=

⋁

{

a

n

⁢

b

∣

n

∈

ℕ

∪

{

0

}

}

≤

b

.

–

similarly, if

b

⁢

a

≤

b

, then

b

⁢

a

*

≤

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:

a

⁢

a

=

a

for any

a

∈

Q

. If

⋅

is idempotent and

1

=

1

′

, then

⋅

⁣

=

⁣

∧

. In other words,

a

⁢

b

=

a

∧

b

.

Proof.

As we have seen,

a

⁢

b

≤

a

∧

b

in the 2 above. Now, suppose

c

≤

a

∧

b

. Then

c

≤

a

and

c

≤

b

, so

c

=

c

2

≤

c

⁢

b

≤

a

⁢

b

. So

a

⁢

b

is the

greatest lower bound

of

a

and

b

, i.e.,

a

⁢

b

=

a

∧

b

. This also means that

b

⁢

a

=

b

∧

a

=

a

∧

b

=

a

⁢

b

.
∎

7.

In fact, a locale is a quantale if we define

⋅

⁣

:=

⁣

∧

. Conversely, a quantale where

⋅

is idempotent and

1

=

1

′

is a locale.

Proof.

If

Q

is a locale with

⋅

⁣

=

⁣

∧

, then

a

⁢

a

=

a

∧

a

=

a

and

a

⁢

1

=

a

∧

1

=

a

=

1

∧

a

=

1

⁢

a

, implying

1

=

1

′

. The

infinite

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

)

=

⋁

{

a

⁢

s

∣

s

∈

S

}

=

⋁

{

a

∧

s

∣

s

∈

S

}

. Similarly

(

⋁

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).

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

Generated on Fri Feb 9 10:05:32 2018 by

LaTeXML