background image

14.1. ORDERING OF FILTERS

241

Proposition

1285

.

For ultrafilters

2

is the same as Rudin-Keisler ordering

(as defined in [

40

]).

Proof.

x

2

y

iff there exist sets

A

x

and

B

y

and a bijective

Set

-

morphism

f

:

X

Y

such that

y

÷

B

=

C

P

Y

h

f

1

i

C

x

÷

A

that is when

C

y

÷

B

f

1

C

x

÷

A

what is equivalent to

C

y

f

1

C

x

what is the definition of Rudin-Keisler ordering.

Remark

1286

.

The relation of being isomorphic for ultrafilters is traditionally

called

Rudin-Keisler equivalence

.

Obvious

1287

.

(

1

)

(

2

).

Definition

1288

.

Let

Q

and

R

be binary relations on the set of (small) filters.

I will denote

MonRld

Q,R

the directed multigraph with objects being filters 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

filters and morphisms 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.

Lemma

1289

.

CoMonRld

Q,R

6

=

∅ ⇔

MonRld

Q,R

6

=

.

Proof.

f

CoMonRld

Q,R

(im

f

)

Q

A ∧

(dom

f

)

R

B ⇔

(dom

f

1

)

Q

A ∧

(im

f

1

)

R

B ⇔

f

1

MonRld

Q,R

for every monovalued reloid

f

(or what is

the same, injective reloid

f

1

).

Theorem

1290

.

For every filters

A

and

B

the following are equivalent:

1

.

A ≥

1

B

.

2

. Hom

MonRld

=

,

w

(

A

,

B

)

6

=

.

3

. Hom

MonRld

v

,

w

(

A

,

B

)

6

=

.

4

. Hom

MonRld

v

,

=

(

A

,

B

)

6

=

.

5

. Hom

CoMonRld

=

,

w

(

A

,

B

)

6

=

.

6

. Hom

CoMonRld

v

,

w

(

A

,

B

)

6

=

.

7

. Hom

CoMonRld

v

,

=

(

A

,

B

)

6

=

.

Proof.

1

2

There exists a

Set

-morphism

f

: Base(

A

)

Base(

B

) such that

B v

FCD

f

A

. We have

dom(

RLD

f

)

|

A

=

A u >

(Base(

A

)) =

A

and

im(

RLD

f

)

|

A

= im(

FCD

)(

RLD

f

)

|

A

= im(

FCD

f

)

|

A

=

FCD

f

A w B

.

Thus (

RLD

f

)

|

A

is a monovalued reloid such that dom(

RLD

f

)

|

A

=

A

and im(

RLD

f

)

|

A

w B

.

2

3

,

4

3

,

5

6

,

7

6

Obvious.

3

1

We have

B

v

h

(

FCD

)

f

iA

for a monovalued reloid

f

RLD

(Base(

A

)

,

Base(

B

)).

Then there exists a

Set

-morphism

F

: Base(

A

)

Base(

B

) such that

B v

FCD

F

A

that is

A ≥

1

B

.