extension of a poset

extension of a poset

Let

(

P

,

≤

P

)

be a poset. An

extension

of

(

P

,

≤

P

)

is a poset

(

Q

,

≤

Q

)

such that

•

Q

=

P

, and

•

if

a

≤

P

b

, then

a

≤

Q

b

.

From the

definition

, the underlying set of an extension of a poset does not change. We may therefore, when there is no ambiguity, say that

Q

is an extension of a poset

P

and use

≤

Q

and

≤

P

to distinguish the

partial ordering

on

Q

and

P

respectively.

Every poset has a

trivial

extension, namely itself. A poset is

maximal

if the only extension is the trivial one. Given a poset that is not maximal, can we find a non-trivial extension? Suppose

P

is a poset and

(

a

,

b

)

∉

≤

P

(

a

,

b

not

comparable

). Then

≤

Q

:=

≤

P

∪

A

, where

A

=

{

(

r

,

s

)

∣

r

≤

a

⁢

and

⁢

b

≤

s

}

, is a non-trivial

partial order

extending

≤

P

, non-trivial since

(

a

,

b

)

∈

≤

Q

-

≤

P

. So a

maximal poset

is just a

linearly ordered set

, as any pair of

elements

is comparable.

An extension

Q

of

P

is said to be

linear

if

Q

is a linearly ordered set. By Zorn’s lemma, and the construction above, every poset

P

has a

linear extension

: take

𝒫

to be the poset of all extension of

P

(ordered by

inclusion

), given a chain

𝒞

of elements in

𝒫

, the union

⋃

𝒞

is an element of

𝒫

, so that

𝒫

has a

maximal element

, which can easily be seen to be linear. Thus we have just proved what is known as the

order extension principle

:

Proposition 1

(order extension principle)

Every partial ordering on a set can be extended to a

linear ordering

.

And since every set is trivially a poset, where

a

≤

b

iff

a

=

b

, we record the following

corollary

, known as the

ordering principle

:

Corollary 1

(ordering principle)

Every set can be linearly ordered.

In fact, every poset is the

intersection

of all of its linear extensions

P

=

⋂

{

L

∣

L

⁢

is a linear extension of

⁢

P

}

,

for if

(

a

,

b

)

is not in the intersection, a linear extension

M

of

P

can be constructed via above containing

(

a

,

b

)

, which is a

contradiction

. A set

𝒮

of linear extensions of a poset

P

is said to be a

realizer

of

P

if

⋂

𝒮

=

P

. Every poset has a realizer.

Remark

. Instead of the stronger

axiom of choice

, the order extension principle can be proved using the weaker

Boolean prime ideal theorem

. Furthermore, the ordering principle implies the

axiom of choice for finite sets

.

References

1

W. T. Trotter,

Combinatorics and

Partially Ordered Sets

, Johns-Hopkins University Press, Baltimore (1992).

2

T. J. Jech,

The Axiom of Choice

, North-Holland Pub. Co., Amsterdam, (1973).

Title

extension of a poset

Canonical

name

ExtensionOfAPoset

Date of creation

2013-03-22 16:31:57

Last modified on

2013-03-22 16:31:57

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06A06

Synonym

linear extension

Defines

maximal poset

Defines

linear extension of a poset

Defines

realizer

Defines

ordering principle

Defines

order extension principle

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