properties of well-ordered sets

properties of well-ordered sets

The purpose of this entry is to collect

properties

of

well-ordered
sets

. We denote all

orderings

uniformly by

≤

. If you are
interested in history, have a look at

[

C

]

.

The following properties are easy to see:

•

If

A

is a

totally ordered set

such that for every subset

B

⊆

A

the set of all elements of

A

strictly

greater than
the elements of

B

,

B

>

:=

{

a

∈

A

∖

B

∣

b

≤

a

⁢

for all

⁢

b

∈

B

}

,

is either empty or has a least

element

, then

A

is well-ordered.

•

Every subset of a well-ordered set is well-ordered.

•

If

A

is well-ordered and

B

is a poset such that there is a

bijective

order morphism

φ

:

A

→

B

, then

B

is
well-ordered.

Now we define an important ingredient for understanding the

structure

of well-ordered sets.

Definition 1

(section).

Let

A

be well-ordered. Then for every

a

∈

A

we define the

section

of

a

:

a

^

:=

{

b

∈

A

∣

b

≤

a

}

.

A section is also known as an

initial segment

. We denote the set of all sections of

A

by

A

^

. This set is
ordered by

inclusion

.

Theorem 1

.

Let

A

be a well-ordered set. Then the

mapping

⋅

^

:

A

→

A

^

defined by

a

↦

a

^

is a bijective order morphism.
In particular,

A

^

is well-ordered.

Proof.

Let

a

,

b

∈

A

with

a

≤

b

. Then

a

^

⊆

b

^

, so

⋅

^

is an order morphism. Now assume that

a

^

=

b

^

. If

a

didn’t equal

b

, we would have

b

∉

a

^

, leading to a

contradiction

. Therefore

⋅

^

is

injective

. Now let

C

∈

A

^

, then there exists a

c

∈

A

such that

c

^

=

C

, so

⋅

^

is

surjective

.
∎

Theorem 2

.

Let

A

and

B

be well-ordered sets and

φ

:

A

→

B

a
bijective order morphism. Then there exists a bijective order morphism

φ

^

:

A

^

→

B

^

such that for all

a

∈

A

φ

^

⁢

(

a

^

)

=

φ

⁢

(

a

)

^

.

Proof.

Setting

φ

^

⁢

(

a

^

)

:=

φ

⁢

(

a

)

^

is

well-defined

by

Theorem

1

. The rest of the theorem follows
since

⋅

^

and

φ

are bijective order morphisms.
∎

Theorem 3

.

Let

A

be a well-ordered set and

a

∈

A

such that there is an
injective order morphism

φ

:

A

→

a

^

. Then

A

=

a

^

.

Proof.

The image of a section of

A

under

φ

has a

maximal element

which in turn defines a smaller section of

A

. We may therefore
define the following two monotonically decreasing

sequences

of sets:

B

0

:=

φ

⁢

(

a

^

)

,

A

0

:=

section defined by maximal element of

⁢

B

0

,

B

n

:=

φ

⁢

(

A

n

-

1

)

,

A

n

:=

section defined by maximal element
of

⁢

B

n

.

Now

A

^

is well-ordered, so the set defined by the elements of the
sequence

(

A

n

)

has a minimal element, that is

A

N

=

A

N

+

1

and
hence

B

N

=

B

N

+

1

for some sufficiently large

N

. Applying

φ

-

1

N

+

1

times to the latter equation yields

a

^

=

A

0

,
that is

a

is the maximal element of

B

, and thus of

A

.
∎

Theorem 4

.

Let

A

and

B

be well-ordered sets. Then there exists at most one
bijective order morphism

φ

:

A

→

B

.

Proof.

Let

φ

,

ψ

:

A

→

B

be two bijective order morphisms. Let

a

∈

A

and set

b

:=

φ

⁢

(

a

)

and

c

:=

ψ

⁢

(

a

)

. Then the

restrictions

φ

|

a

^

:

a

^

→

b

^

and

ψ

|

a

^

:

a

^

→

c

^

are bijective order
morphisms, so the restriction of

ψ

⁢

φ

-

1

to

b

^

is a
bijective order morphism to

c

^

. Now either

b

^

⊆

c

^

or

c

^

⊆

b

^

, so by Theorem

3

b

^

=

c

^

, hence

b

=

c

and thus

φ

=

ψ

.
∎

Theorem 5

.

Let

A

and

B

be well-ordered sets such that for every section

a

^

∈

A

^

there is a bijective order morphism to a section

b

^

∈

B

^

and vice-versa, then there is a bijective order
morphism

φ

:

A

→

B

.

Proof.

Let

a

∈

A

and let

b

^

∈

B

^

be a section such that there is a
bijective order morphism

ψ

a

:

a

^

→

b

^

. By
Theorem

3

,

b

^

is unique, and so is

ψ

a

by Theorem

4

. Defining

φ

:

A

→

B

by setting

φ

⁢

(

a

)

=

b

gives therefore a
well-defined (by Theorem

1

) and injective order
morphism. But

φ

is also surjective, since any

b

∈

B

maps
uniquely to

A

via

b

^

→

a

^

, and back again by

φ

.
∎

Theorem 6

.

Let

A

and

B

be well-ordered sets. Then there is an injective order
morphism

ι

:

A

→

B

or

ι

:

B

→

A

. If

ι

cannot be chosen bijective, then it can at least be chosen such that
its image is a section.

Proof.

Let

A

^

0

be the set of sections of

A

from which there is an
injective order morphism to

B

. If

A

^

0

is the

empty set

, then

B

must be empty, since otherwise we could map the least element of

A

to

B

. If

A

^

0

is not empty, we may consider the set

A

0

:=

∩

A

^

0

. If

A

0

=

A

, nothing remains to be shown. Otherwise
the set

A

∖

A

0

is nonempty an hence has a least element

a

0

∈

A

. By construction, there is no injective order morphism from

a

^

0

to

B

, but there is an injective order morphism from

φ

a

:

a

^

→

B

for every element

a

∈

A

which is
strictly smaller than

a

0

. Now assume there is an element

b

∈

B

such that there is no injective order morphism from

b

^

→

A

. Then we can similarly construct a least element

b

0

∈

B

for
which there is no injective order morphism

b

^

0

→

A

. Surely,

b

0

is greater than all the elements from the images of the

functions

φ

a

, but then there is a bijective order
morphism from

a

^

0

to

b

^

0

by
Theorem

5

which is a
contradiction. Therefore, all sections of

B

and

B

itself
map injectively and

order-preserving

to

A

.
∎

Theorem 7

.

Let

A

be a well-ordered set and

B

⊆

A

a nonempty
subset. Then there is a bijective order morphism from

B

to one of
the sets in

A

^

∪

{

A

}

.

Proof.

The set

B

is well-ordered with respect to the order induced by

A

. Assume a bijective order morphism as stated by the theorem does
not exist. Then, by virtue of Theorem

6

, there is
an injective but not surjective order morphism

ι

:

A

→

B

whose
image is a section

b

^

∈

B

^

. The element

b

defines a section
in

A

^

which is identical to

A

by
Theorem

3

. Thus

ι

is surjective which is
a contradiction.
∎

References

C

G. Cantor

, Beiträge zur Begründung der
transfiniten Mengenlehre (Zweiter Artikel),

Math. Ann.

49,
207–246 (1897).

Title

properties of well-ordered sets

Canonical

name

PropertiesOfWellorderedSets

Date of creation

2013-03-22 15:23:42

Last modified on

2013-03-22 15:23:42

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

10

Author

GrafZahl (9234)

Entry type

Theorem

Classification

msc 06A05

Synonym

initial segment

Defines

section

Generated on Fri Feb 9 09:50:29 2018 by

LaTeXML