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Remark 39.

The relation of being isomorphic for ultrafilters is traditionally called

Rudin-Keisler

equivalence

.

Obvious 40.

(

>

1

)

(

>

2

)

.

Definition 41.

Let

Q

and

R

are binary relations on the set of filter objects. I will denote

MonRld

Q,R

the directed multigraph with objects being filter objects and morphisms such mono-

valued 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 filter objects 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.

Theorem 42.

For every f.o.

A

and

B

the following are equivalent:

1.

A

>

1

B

.

2. Mor

MonRld

=

,

(

A

;

B

)

.

3. Mor

MonRld

,

(

A

;

B

)

.

4. Mor

MonRld

,

=

(

A

;

B

)

.

5. Mor

CoMonRld

=

,

(

A

;

B

)

.

6. Mor

CoMonRld

,

(

A

;

B

)

.

7. Mor

CoMonRld

,

=

(

A

;

B

)

.

Proof.

(1)

(2).

There exists a

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

such that

B ⊆ h↑

f

iA

. We have

dom

(

RLD

f

)

|

A

=

A ∩

dom

f

=

A

and

im

(

RLD

f

)

|

A

=

im

(

FCD

)(

RLD

f

)

|

A

=

im

(

f

)

|

A

=

h↑

f

iA ⊇ B

.

Thus

(

RLD

f

)

|

A

is a monovalued reloid such that dom

(

RLD

f

)

|

A

=

A

and im

(

RLD

f

)

|

A

⊇B

.

(2)

(3), (4)

(3), (5)

(6), (7)

(6).

Obvious.

(3)

(1).

We have

B ⊆ 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 ⊆ h↑

F

iA

that is

A

>

1

B

.

(6)

(7).

dom

f

|

B

=

B

and im

f

|

B

⊆A

.

(2)

(5), (3)

(6), (4)

(7).

By duality.

Theorem 43.

For every f.o.

A

and

B

the following are equivalent:

1.

A

>

2

B

.

2. Mor

MonRld

=

,

=

(

A

;

B

)

.

3. Mor

CoMonRld

=

,

=

(

A

;

B

)

.

Proof.

(1)

(2).

Let

A

>

2

B

that is

B

=

h↑

f

iA

for some

Set

-morphism

f

:

Base

(

A

)

Base

(

B

)

.

Then dom

(

RLD

f

)

|

A

=

A

and im

(

RLD

f

)

|

A

=

im

(

FCD

)(

RLD

f

)

|

A

=

im

(

f

)

|

A

=

h↑

f

iA

=

B

. So

(

RLD

f

)

|

A

is a sought for reloid.

(2)

(1).

There exists a

Set

-morphism

F

:

Base

(

A

)

Base

(

B

)

such that

f

= (

RLD

F

)

|

A

. Thus

h↑

F

iA

=

im

(

F

)

|

A

=

im

(

FCD

)(

RLD

F

)

|

A

=

im

(

FCD

)

f

=

im

f

=

B

. Thus

A

>

2

B

is testified by

the morphism

F

.

(2)

(3).

By duality.

Ordering of filters

7