ideal completion of a poset

ideal completion of a poset

Let

P

be a poset. Consider the set

Id

⁡

(

P

)

of all

order ideals

of

P

.

Theorem 1

.

Id

⁡

(

P

)

is an

algebraic dcpo

, such that

P

can be embedded in.

Proof.

We shall list, and when necessary, prove the following series of facts which ultimately prove the main assertion. For convenience, write

P

′

=

Id

⁡

(

P

)

.

1.

P

′

is a poset with

≤

defined by set theoretic

inclusion

.

2.

For any

x

∈

P

,

↓

x

∈

P

′

.

3.

P

can be embedded in

P

′

. The

function

f

:

P

→

P

′

defined by

f

(

x

)

=

↓

x

is order preserving and

one-to-one

. If

x

≤

y

, and

a

≤

x

, then

a

≤

y

, hence

↓

x

⊆

↓

y

. If

↓

x

=

↓

y

, we have that

x

≤

y

and

y

≤

x

, so

x

=

y

, since

≤

is antisymmetric.

4.

P

′

is a dcpo. Suppose

D

is a

directed set

in

P

′

. Let

E

=

⋃

D

. For any

x

,

y

∈

E

,

x

∈

I

and

y

∈

J

for some ideals

I

,

J

∈

D

. As

D

is directed, there is

K

∈

D

such that

I

⊆

K

and

J

⊆

K

. So

x

,

y

∈

K

and hence there is

z

∈

K

⊆

E

such that

x

≤

z

and

y

≤

z

. This shows that

E

is directed. Next, suppose

x

∈

E

and

y

≤

x

. Then

x

∈

I

for some

I

∈

D

, so

y

∈

I

⊆

E

as well. This shows that

E

is a down set. So

E

is an ideal of

P

:

⋁

D

=

E

∈

P

′

.

5.

For every

x

∈

P

,

↓

x

is a

compact element

of

P

′

. If

↓

x

≤

⋁

D

, where

D

is directed in

P

′

, then

↓

x

⊆

⋃

D

, or

x

∈

⋃

D

, which implies

x

∈

I

for some ideal

I

∈

D

. Therefore

↓

x

⊆

I

, and

↓

x

is way below itself:

↓

x

is compact.

6.

P

′

is an algebraic dcpo. Let

I

∈

P

′

. Let

C

=

{

↓

x

∣

x

∈

I

}

. For any

x

,

y

∈

I

, there is

z

∈

I

such that

x

≤

z

and

y

≤

z

. This shows that

↓

x

≤

↓

z

and

↓

y

≤

↓

z

in

C

, so that

C

is directed. It is easy to see that

I

=

⋁

C

. Since

I

is a join of a directed set consisting of compact elements,

P

′

is algebraic.

This

completes

the proof.
∎

Definition

.

Id

⁡

(

P

)

is called the

ideal completion

of

P

.

Remarks

.

•

In general, the ideal completion of a poset is not a

complete lattice

. It is complete in the sense of being

directed complete

. This is different from another type of completion, called the

MacNeille completion

of

P

, which is a complete lattice.

•

If

P

is an

upper semilattice

, then so is

Id

⁡

(

P

)

. In fact, the join of any non-empty family of ideals exists. Furthermore, if

P

has a bottom

element

0

, then

Id

⁡

(

P

)

is a complete lattice.

Proof.

Let

S

be a non-empty family of ideals in

P

. Let

A

be the set of

P

consisting of all finite joins of elements of those ideals in

S

, and

B

=

↓

A

. Clearly,

B

is a

lower set

. For every

a

,

b

∈

B

, we have

c

,

d

∈

A

such that

a

≤

c

and

b

≤

d

. Since

c

and

d

are both finite joins of elements of those ideals in

S

, so is

c

∨

d

. Since

a

≤

c

∨

d

and

b

≤

c

∨

d

,

B

is directed. If

I

is any ideal larger than any of the ideals in

S

, clearly

A

⊆

I

, since

I

is directed. So

B

=

↓

A

⊆

↓

I

=

I

. Therefore,

B

=

⋁

S

.

If

0

∈

P

, then

⟨

0

⟩

, the bottom of

Id

⁡

(

P

)

, is the join of the empty family of ideals in

P

. By this entry (

http://planetmath.org/CriteriaForAPosetToBeACompleteLattice

),

Id

⁡

(

P

)

is a complete lattice.
∎

•

If

P

is a

lower semilattice

, then so is

Id

⁡

(

P

)

.

Proof.

Let

I

,

J

be two ideals in

P

and

K

=

I

∩

J

. By definition,

I

and

J

are non-empty, so let

a

∈

I

and

b

∈

J

. As

P

is a lower semilattice,

c

:=

a

∧

b

exists and

c

≤

a

and

c

≤

b

. So

c

∈

I

∩

J

, and that

K

=

I

∩

J

is non-empty. If

x

≤

y

∈

K

, then

x

≤

y

∈

I

or

x

∈

I

. Similarly

x

∈

J

. Therefore

x

∈

I

∩

J

=

K

and

K

is a lower set. If

r

,

s

∈

K

, then there is

u

∈

I

and

v

∈

J

such that

r

,

s

≤

u

,

v

. So

r

,

s

≤

u

∧

v

and

K

is directed. This means that

I

∩

J

∈

Id

⁡

(

P

)

.
∎

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

ideal completion of a poset

Canonical name

IdealCompletionOfAPoset

Date of creation

2013-03-22 17:03:01

Last modified on

2013-03-22 17:03:01

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A12

Classification

msc 06A06

Related topic

LatticeOfIdeals

Defines

ideal completion

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