example of a universal structure

example of a universal structure

Let

L

be the

first order language

with the

binary relation

≤

. Consider the following

sentences

:

•

∀

x

,

y

(

(

x

≤

y

∨

y

≤

x

)

∧

(

(

x

≤

y

∧

y

≤

x

)

↔

x

=

y

)

)

•

∀

x

,

y

,

z

(

x

≤

y

∧

y

≤

z

→

x

≤

z

)

Any

L

-

structure

satisfying these is called a

linear order

.
We define the

relation

<

so that

x

<

y

iff

x

≤

y

∧

x

≠

y

. Now consider these sentences:

1.

∀

x

,

y

(

(

x

<

y

→

∃

z

(

x

<

z

<

y

)

)

2.

∀

x

∃

y

,

z

(

y

<

x

<

z

)

A linear order that

satisfies

1. is called dense (

http://planetmath.org/DenseTotalOrder

).
We say that a linear order that satisfies 2. is

without

endpoints

.
Let

T

be the theory of

dense linear orders

without endpoints. This is a

complete theory

.

We can see that

(

ℚ

,

≤

)

is a model of

T

. It is actually a rather special model.

Theorem 1

Let

(

S

,

≤

)

be any finite linear order. Then

S

embeds in

(

Q

,

≤

)

.

Proof: By

induction

on

|

S

|

, it is

trivial

for

|

S

|

=

1

.

Suppose that the statement holds for all linear orders with

cardinality

less than or equal to

n

.
Let

|

S

|

=

n

+

1

, then pick some

a

∈

S

, let

S

′

be the structure induced by

S

on

S

∖

a

.
Then there is some

embedding

e

of

S

′

into

ℚ

.

•

Now suppose

a

is less than every member of

S

′

, then as

ℚ

is without endpoints, there is some

element

b

less than every element in the image of

e

.
Thus we can extend

e

to map

a

to

b

which is an embedding of

S

into

ℚ

.

•

We work similarly if

a

is greater than every element in

S

′

.

•

If neither of the above hold then we can pick some maximum

c

1

∈

S

′

so that

c

1

<

a

.
Similarly we can pick some minimum

c

2

∈

S

′

so that

c

2

<

a

. Now there is some

b

∈

ℚ

with

e

⁢

(

c

1

)

<

b

<

e

⁢

(

c

2

)

. Then extending

e

by

mapping

a

to

b

is the required embedding.

□

It is easy to extend the above result to

countable

structures.
One views a countable structure as a the union of an increasing chain of finite

substructures

.
The necessary embedding is the union of the embeddings of the substructures. Thus

(

ℚ

,

≤

)

is

universal

countable linear order.

Theorem 2

(

ℚ

,

≤

)

is

homogeneous

.

Proof: The following type of proof is known as a

back and forth

argument

.
Let

S

1

and

S

2

be two finite substructures of

(

ℚ

,

≤

)

.
Let

e

:

S

1

→

S

2

be an

isomorphism

.
It is easier to think of two

disjoint

copies

B

and

C

of

ℚ

with

S

1

a substructure of

B

and

S

2

a substructure of

C

.

Let

b

1

,

b

2

,

…

be an enumeration of

B

∖

S

1

.
Let

c

1

,

c

2

,

…

,

c

n

be an enumeration of

C

∖

S

2

. We iterate the following two step process:

The

i

th forth step

If

b

i

is already in the domain of

e

then do nothing. If

b

i

is not in the domain of

e

.
Then as in

proposition

1

, either

b

i

is less than every element in the domain of

e

or greater than or it has an immediate

successor

and

predecessor

in the range of

e

.
Either way there is an element

c

in

C

∖

range

⁡

(

e

)

relative to the range of

e

.
Thus we can extend the isomorphism to include

b

i

.

The

i

th back step

If

c

i

is already in the range of

e

then do nothing. If

c

i

is not in the domain of

e

.
Then exactly as above we can find some

b

∈

B

∖

dom

⁡

(

e

)

and extend

e

so that

e

⁢

(

b

)

=

c

i

.

After

ω

stages, we have an isomorphism whose range includes every

b

i

and whose domain includes every

c

i

. Thus we have an isomorphism from

B

to

C

extending

e

.

□

A

similar

back and forth argument shows that any countable dense linear order without endpoints is isomorphic to

(

ℚ

,

≤

)

so

T

is

ℵ

0

-

categorical

.

Title

example of a universal structure

Canonical name

ExampleOfAUniversalStructure

Date of creation

2013-03-22 13:23:16

Last modified on

2013-03-22 13:23:16

Owner

uzeromay (4983)

Last modified by

uzeromay (4983)

Numerical id

15

Author

uzeromay (4983)

Entry type

Example

Classification

msc 03C50

Classification

msc 03C52

Related topic

Homogeneous4

Related topic

KappaCategorical

Related topic

DifferentialEquation

Related topic

ExampleOfDefinableType

Related topic

RandomGraph

Defines

back and forth

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