complete distributivity

complete distributivity

A

lattice

L

is said to be

completely distributive

if it is a

complete lattice

such that, given any sets

K

⊆

I

×

J

such that

K

projects onto

I

, and any subset

{

x

i

⁢

j

∣

(

i

,

j

)

∈

K

}

of

L

,

⋀

i

∈

I

(

⋁

j

∈

K

⁢

(

i

)

x

i

⁢

j

)

=

⋁

f

∈

A

(

⋀

i

∈

I

x

i

⁢

f

⁢

(

i

)

)

,

(1)

where

K

⁢

(

i

)

:=

{

j

∈

J

∣

(

i

,

j

)

∈

K

}

, and

A

=

{

f

:

I

→

J

∣

f

(

i

)

∈

K

(

i

)

for all

i

∈

I

}

.

By setting

I

=

J

=

{

1

,

2

}

and

K

=

{

(

1

,

1

)

,

(

2

,

1

)

,

(

2

,

2

)

}

, then

K

⁢

(

1

)

=

{

1

}

,

K

⁢

(

2

)

=

{

1

,

2

}

, and

A

consists of two

functions

{

(

1

,

1

)

,

(

2

,

1

)

}

and

{

(

1

,

1

)

,

(

2

,

2

)

}

. Then, the equation above reads:

(

x

11

)

∧

(

x

21

∨

x

22

)

=

(

x

11

∧

x

21

)

∨

(

x

11

∧

x

22

)

which is one of the

distributive laws

, so that complete distributivity implies distributivity.

More generally, setting

I

=

{

1

,

2

}

and

J

containing

1

but otherwise arbitrary, and

K

=

{

(

2

,

j

)

∣

j

∈

J

}

∪

{

(

1

,

1

)

}

. Then

K

⁢

(

1

)

=

{

1

}

,

K

⁢

(

2

)

=

J

, and

A

is the set of functions from

I

to

J

fixing

1

, and the equation (1) above now looks like

x

11

∧

(

⋁

{

x

2

⁢

j

∣

j

∈

J

}

=

⋁

{

x

11

∧

x

2

⁢

j

∣

j

∈

J

}

which shows that completely distributivity implies

join

infinite

distributivity (

http://planetmath.org/JoinInfiniteDistributive

)

.

Remarks.

1.

Dualizing the above equation results in the same lattice. In other words, a completely distributive lattice may be equivalently defined using the dual of Equation (1). As a result, a completely distributive lattice also

satisfies

MID, and hence is

infinite distributive

.

2.

However, a

complete

distributive lattice

does not have to be completely distributive. Here’s an example: let

ℕ

be the set of natural numbers with the usual

ordering

, and

ℕ

′

be an identical copy of

ℕ

such that each

natural number

n

corresponds to

n

′

∈

ℕ

′

. Then

ℕ

′

has a natural ordering

induced

by the usual ordering on

ℕ

. Take the union

N

of these two sets. Then

N

becomes a lattice if we extend the meets and joins on

ℕ

and

ℕ

′

by additionally setting

a

′

∨

b

:=

{

a

′

if

⁢

b

≤

a

,

b

′

otherwise.

and

a

′

∧

b

:=

{

b

if

⁢

b

≤

a

,

a

otherwise.

Finally, let

L

be the lattice formed from

N

by adjoining an extra

element

∞

to be its top element. It is not hard to see that

L

is complete and distributive. However,

L

is not completely distributive, for

0

′

∧

(

⋁

{

a

∣

a

∈

ℕ

}

)

=

0

′

∧

∞

=

0

′

, whereas

⋁

{

0

′

∧

a

∣

a

∈

ℕ

}

=

⋁

{

0

}

=

0

≠

0

′

.

3.

In some literature, completeness

assumption

is not required, so that the equation (1) above is conditionally defined. In other words, the equation is defined only when each side of the equation exists first.

4.

Another

generalization

is the so-called

(

𝔪

,

𝔫

)

-distributivity, where

𝔪

and

𝔫

are

cardinal numbers

. Specifically, a lattice

L

is

(

𝔪

,

𝔫

)

-distributive if it is complete and equation (1) is true whenever

I

has

cardinality

≤

𝔪

and each

K

⁢

(

i

)

has cardinality

≤

𝔫

for each

i

∈

I

.

References

1

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott,

Continuous

Lattices and Domains

, Cambridge University Press, Cambridge (2003).

Title

complete distributivity

Canonical name

CompleteDistributivity

Date of creation

2013-03-22 15:41:34

Last modified on

2013-03-22 15:41:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

22

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06D10

Related topic

JoinInfiniteDistributive

Defines

completely distributive

Defines

Defines

n)-distributive

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