domain

domain

A poset

P

is said to be a

directed complete partially ordered set

, or

dcpo

for short, if every

directed set

D

⊆

P

has a

supremum

. Since the

empty set

is directed, a dcpo must have a bottom

element

.

A

domain

P

is a

continuous

dcpo. Here, continuous means that

P

is a

continuous poset

.

Example

. Let

A

,

B

be sets. Consider the set

P

of all

partial functions

from

A

to

B

. This means that any

f

∈

P

is a

function

C

→

B

, for some subset

C

of

A

. We show that

P

is a domain.

1.

P

is a poset: Define a

binary relation

on

P

as follows:

f

≤

g

iff

g

is an

extension

of

f

. In other words, if

f

:

C

→

B

and

g

:

D

→

B

, then

C

⊆

D

and

f

⁢

(

x

)

=

g

⁢

(

x

)

for all

x

∈

C

. Clearly,

≤

is

reflexive

, anti-symmetric, and

transitive

. So

≤

turns

P

into a poset.

2.

P

is a dcpo: Suppose that

D

is a directed subset of

P

. Set

E

=

⋃

{

dom

⁡

(

f

)

∣

f

∈

D

}

. Define

g

:

E

→

B

as follows: for any

x

∈

E

,

g

⁢

(

x

)

=

f

⁢

(

x

)

where

x

∈

dom

⁡

(

f

)

for some

f

∈

D

. Is this well-defined? Suppose

x

∈

dom

⁡

(

f

1

)

∩

dom

⁡

(

f

2

)

. Since

D

is directed, there is an

f

∈

D

extending both

f

1

and

f

2

. This means that

f

1

⁢

(

x

)

=

f

⁢

(

x

)

=

f

2

⁢

(

x

)

. Therefore,

g

:=

⋁

D

is a well-defined function (on

E

). Hence

P

is a dcpo.

3.

If

f

,

g

≪

h

, then

f

∨

g

≪

h

: Since

h

extends both

f

and

g

,

a

:=

f

∨

g

:

dom

⁡

(

f

)

∪

dom

⁡

(

g

)

→

B

is well-defined (the construction is the same as above). To show that

a

≪

h

, suppose

D

⊆

P

is directed and

h

≤

⋁

D

(note that

⋁

D

exists by 2 above). Since

f

≪

h

, there is

r

∈

D

such that

f

≤

r

. Similarly,

g

≪

h

implies an

s

∈

D

with

g

≤

s

. Since

D

is directed, there is

t

∈

D

with

r

,

s

≤

t

. This means

f

≤

t

and

g

≤

t

, or

a

=

f

∨

g

≤

t

.

4.

P

is continuous: Let

wb

⁡

(

h

)

=

{

f

∈

P

∣

f

≪

h

}

. Then by 3 above,

wb

⁡

(

h

)

is a directed set. By 2,

b

:=

⋁

wb

⁡

(

h

)

exists, and

b

≤

h

. Suppose

x

∈

dom

⁡

(

h

)

. Then the function

c

x

:

{

x

}

→

B

defined by

c

x

⁢

(

x

)

=

h

⁢

(

x

)

is way below

h

, for if

h

≤

⋁

D

, then

x

∈

dom

⁡

(

f

)

for some

f

∈

D

, or

dom

⁡

(

c

x

)

=

{

x

}

⊆

dom

⁡

(

f

)

, which means

c

x

≤

f

. Therefore,

c

x

≤

b

. This implies that

dom

⁡

(

h

)

=

⋁

{

dom

⁡

(

c

x

)

∣

x

∈

dom

⁡

(

h

)

}

⊆

dom

⁡

(

b

)

. As a result,

h

≤

b

.

Remark

. Domain theory is a branch of order theory that is used extensively in theoretical computer science. As in the example, one can think of a domain as a

collection

of partial pictures or pieces of partial information. Being continuous is the same as saying that any picture or piece of information can be pieced together by partial ones by way of “approximations”.

Title

domain

Canonical name

Domain12

Date of creation

2013-03-22 16:49:25

Last modified on

2013-03-22 16:49:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06B35

Synonym

directed complete

Synonym

directed complete poset

Synonym

directed complete partially ordered set

Related topic

CompleteLattice

Defines

domain

Defines

dcpo

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