background image

If card

(

X

n

A

) =

card

(

Y

n

B

)

our statement follows from the last lemma.

Now assume without loss of generality card

(

X

n

A

)

<

card

(

Y

n

B

)

.

card

B

=

card

Y

because card

(

Y

n

B

)

<

card

Y

.

It is easy to show that there exists

B

0

B

such that card

(

X

n

A

) =

card

(

Y

n

B

0

)

and

card

B

0

=

card

B

.

We will nd a bijection

g

from

B

to

B

0

which witnesses direct isomorphism of

v

to

v

itself.

Then the composition

g

f

witnesses a direct isomorphism of

u

A

and

v

B

0

and by the lemma

u

and

v

are directly isomorphic.

Let

D

=

B

0

n

B

. We have

D

2

/

v

.

There exists a set

E

B

such that card

E

>

card

D

and

E

2

/

v

.

We have card

E

=

card

(

D

[

E

)

and thus there exists a bijection

h

:

E

!

D

[

E

.

Let

g

(

x

) =

x

if

x

2

B

n

E

;

h

(

x

)

if

x

2

E:

g

j

B

n

E

and

g

j

E

are bijections.

im

(

g

j

B

n

E

) =

B

n

E

; im

(

g

j

E

) =

im

h

=

D

[

E

;

(

D

[

E

)

\

(

B

n

E

) = (

D

\

(

B

n

E

))

[

(

E

\

(

B

n

E

)) =

; [ ;

=

;

:

Thus

g

is a bijection from

B

to

(

B

n

E

)

[

(

D

[

E

) =

B

[

D

=

B

0

.

To nish the proof it's enough to show that

h

g

i

v

=

v

. Indeed it follows from

B

n

E

2

v

.

Proposition 13.36.

1. For every

A

2 A

and

B

2 B

we have

A

>

2

B

i

A

>

2

B

.

2. For every

A

2 A

and

B

2 B

we have

A

>

1

B

i

A

>

1

B

.

Proof.

1.

A

>

2

B

i there exist a bijective Set-morphism

f

such that

B

=

h"

FCD

f

iA

. The equality is

obviously preserved replacing

A

with

A

and

B

with

B

.

2.

A

>

1

B

i there exist a bijective Set-morphism

f

such that

B  h"

FCD

f

iA

. The equality is

obviously preserved replacing

A

with

A

and

B

with

B

.

Proposition 13.37.

For ultralters

>

2

is the same as Rudin-Keisler ordering (as dened in [

37

]).

Proof.

x

>

2

y

i there exist sets

A

2

x

and

B

2

y

a bijective Set-morphism

f

:

X

!

Y

such that

y

B

=

f

C

2

P

Y

j h

f

¡

1

i

C

2

x

A

g

that is when

C

2

y

B

, h

f

¡

1

i

C

2

x

A

what is equivalent

to

C

2

y

, h

f

¡

1

i

C

2

x

what is the denition of Rudin-Keisler ordering.

Remark 13.38.

The relation of being isomorphic for ultralters is traditionally called

Rudin-

Keisler equivalence

.

Obvious 13.39.

(

>

1

)

(

>

2

)

.

Denition 13.40.

Let

Q

and

R

be binary relations on the set of lters. I will denote MonRld

Q; R

the directed multigraph with objects being lters and morphisms such monovalued reloids

f

that

(

dom

f

)

Q

A

and

(

im

f

)

R

B

.

I will also denote CoMonRld

Q; R

the directed multigraph with objects being lters and mor-

phisms such injective reloids

f

that

(

im

f

)

Q

A

and

(

dom

f

)

R

B

. These are essentially the duals.

Some of these directed multigraphs are categories with reloid composition (see below). By abuse

of notation I will denote these categories the same as these directed multigraphs.

Theorem 13.41.

For every lters

A

and

B

the following are equivalent:

1.

A

>

1

B

.

2. Mor

MonRld

=

;

w

(

A

;

B

) =

/

;

.

162

Orderings of filters in terms of reloids