chain

chain

Introduction

Let

A

be a poset ordered by

≤

. A subset

B

of

A

is called a

chain

in

A

if any two elements of

B

are

comparable

. In other words, if

a

,

b

∈

B

, then either

a

≤

b

or

b

≤

a

, that is,

≤

is a

total order

on

B

, or that

B

is a

linearly ordered

subset of

A

. When

a

≤

b

, we also write

b

≥

a

. When

a

≤

b

and

a

≠

b

, we write

a

<

b

. When

a

≥

b

and

a

≠

b

, then we write

a

>

b

. A poset is a

chain

if it is a chain as a subset of itself. The

cardinality

of a chain is the cardinality of the underlying set.

Below are some common examples of chains:

1.

𝐧

:=

{

1

,

2

,

…

,

n

}

is a chain under the usual order (

a

≤

b

iff

b

-

a

is non-negative). This is an example of a

finite chain

: a chain whose cardinality is finite. Any

finite set

A

can be made into a chain, since there is a

bijection

from

𝐧

onto

A

, and the total order on

A

is

induced

by the order on

𝐧

. A chain that is not finite is called an

infinite chain

.

2.

ℕ

, the set of natural numbers, is a chain under the usual order. Here we have an example of a

well-ordered set

: a chain such that every non-empty subset has a

minimal element

(as in a poset). Other well-ordered sets are the set

ℚ

+

of

positive

rationals

and

ℝ

+

of positive reals. The

well-ordering principle

states that every set can be well-ordered. It can be shown that the

axiom of choice

is

equivalent

to the well-ordering principle.

3.

ℤ

, the set of integers, is a chain under the usual order.

4.

ℚ

, the set of rationals, is a chain under the usual order. This is an example of a

dense chain

: a chain such that for every pair of distinct

elements

a

<

b

, there is an element

c

such that

a

<

c

<

b

.

5.

ℝ

, the set of reals, is a chain under the usual order. This is an example of a

Dedekind complete

chain

: a chain such that every non-empty

bounded

subset has a

supremum

and an

infimum

.

Constructing chains

One easy way to produce a new chain from an existing one is to form the

dual

of the existing chain: if

A

is a chain, form

A

∂

so that

a

≤

b

in

A

∂

iff

b

≤

a

in

A

.

Another way to produce new chains from existing ones is to form a

join

of chains. Given two chains

A

,

B

, we can form a new chain

A

⁢

∐

B

. The basic idea is to form the

disjoint union

of

A

and

B

, and order this newly constructed set so that the order among elements of

A

is preserved, and similarly for

B

. Furthermore, any element of

A

is always less than any element of

B

(See here (

http://planetmath.org/AlternativeTreatmentOfConcatenation

) for detail).

With these two methods, one can

construct

many more examples of chains:

1.

Take

ℝ

, and form

A

=

ℝ

⁢

∐

{

a

}

. Then

A

is a chain with a top element. If we take

B

=

{

b

}

⁢

∐

A

, we get a chain with both a top and a bottom element. In fact,

B

is an example of a complete chain: a chain such that every subset has a supremum and an infimum. Observe that any finite chain is

complete

.

2.

We can form

ℕ

∂

which is a set with

1

as the top element. We can also form

ℕ

∂

⁢

∐

ℕ

, which has neither top nor bottom, or

ℕ

⁢

∐

ℕ

∂

, which has both a top and a bottom element, but is not complete, as

ℕ

, considered as a subset, has no top. Likewise,

ℕ

∂

is bottomless.

The idea of joining two chains can be generalized. Let

{

A

i

∣

i

∈

I

}

be a family of chains

indexed by

I

, itself a chain. We form

∐

i

∈

I

A

i

as follows: take the disjoint union of

A

i

, which we also write as

∐

i

∈

I

A

i

. Then

(

a

,

i

)

≤

(

b

,

j

)

iff either

i

=

j

and

a

≤

b

, or

i

<

j

.

For example, let

I

=

ℝ

and

A

i

=

ℝ

, with

i

∈

I

. Then

∐

i

∈

I

A

i

is a chain, whose total order is the

lexicographic order

on

ℝ

2

. If we well-order

I

=

ℝ

, then

∐

i

∈

I

A

i

is another chain called the

long line

.

Chain homomorphisms

Let

A

,

B

be chains. A

function

f

from

A

to

B

is said to be a

chain homomorphism

if it is a poset

homomorphism

(it preserves order).

f

⁢

(

A

)

is the homomorphic image of

A

in

B

. Two chains are

homomorphic

if there is a chain homomorphism from one to another. A chain homomorphism is an

embedding

if it is one-to-one. If

A

embeds in

B

, we write

A

⊆

B

. A strict embedding is an embedding that is not onto. If

A

strictly embeds in

B

, we write

A

⊂

B

. An onto embedding is also called an

isomorphism

. If

A

is isomorphic to

B

, we write

A

≅

B

.

Some properties:

•

Two finite chains are isomorphic iff they have the same cardinality.

•

Top and bottom elements are preserved by chain isomorphisms. In other words, if

f

:

A

→

B

is a chain isomorphism and if

a

∈

A

is the top (bottom) element, then

f

⁢

(

a

)

is the top (bottom) element in

B

.

•

In

addition

, the properties of being well-ordered, dense, Dedekind complete, and complete are all preserved under a chain isomorphism.

•

A

⊆

A

⁢

∐

B

. More generally

A

i

⊆

∐

i

∈

I

A

i

.

•

(

A

⁢

∐

B

)

⁢

∐

C

≅

A

⁢

∐

(

B

⁢

∐

C

)

.

•

If

k

is the bottom element of

I

, and

A

k

has a top element

x

, then there is a chain homomorphism

f

:

∐

i

∈

I

A

i

→

A

k

given by

f

⁢

(

a

,

i

)

=

a

if

i

=

k

and

f

⁢

(

a

,

i

)

=

x

if

i

>

k

.

•

Dually, if

k

is the top of

I

and

A

k

has a bottom

x

, then there is a chain homomorphism

f

:

∐

i

∈

I

A

i

→

A

k

given by

f

⁢

(

a

,

i

)

=

a

if

i

=

k

and

f

⁢

(

a

,

i

)

=

x

if

i

<

k

.

•

If

A

k

has both a bottom

x

and a top

y

, then we may define

f

:

∐

i

∈

I

A

i

→

A

k

by

f

⁢

(

a

,

i

)

=

a

if

i

=

k

,

f

⁢

(

a

,

i

)

=

x

if

i

<

k

and

f

⁢

(

a

,

i

)

=

y

if

k

<

i

.

Some examples:

•

𝐧

⊂

ℕ

⊂

ℤ

⊂

ℚ

⊂

ℝ

.

•

𝐧

⁢

∐

𝐦

≅

𝐩

iff

p

=

m

+

n

for any non-negative integers

m

,

n

and

p

.

•

𝐧

⁢

∐

ℕ

≅

ℕ

for any non-negative integer

n

.

•

ℕ

⁢

∐

ℕ

∂

≇

ℕ

∂

⁢

∐

ℕ

.

•

Let

I

be the chain over

ℝ

under the usual order, and

J

the chain over

ℝ

under a well-ordering. Then

∐

i

∈

I

ℝ

≇

∐

j

∈

J

ℝ

.

Title

chain

Canonical name

Chain

Date of creation

2013-03-22 12:09:12

Last modified on

2013-03-22 12:09:12

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03-00

Defines

finite chain

Defines

infinite chain

Defines

dense chain

Defines

complete chain

Defines

join of chains

Defines

chain homomorphism

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