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14.2. RUDIN-KEISLER EQUIVALENCE AND RUDIN-KEISLER ORDER

247

Proof.

Let

f

be a monovalued reloid. There exists a function

F

GR

f

.

Consider the bijective function

p

=

λx

dom

F

: (

x, F x

).

h

p

i

dom

F

=

F

and consequently

h

p

i

dom

f

=

RLD

l

K

up

f

h

p

i

dom

K

=

RLD

l

K

up

f

h

p

i

dom(

K

F

) =

RLD

l

K

up

f

(

K

F

) =

RLD

l

K

up

f

K

=

f.

Thus

p

witnesses that

f

is isomorphic to the filter dom

f

.

Corollary

1310

.

The graph of a monovalued reloid with atomic domain is

atomic.

Corollary

1311

.

id

RLD

A

is isomorphic to

A

for every filter

A

.

Theorem

1312

.

There are atomic filters incomparable by Rudin-Keisler order.

(Elements

a

and

b

are

incomparable

when

a

6v

b

b

6v

a

.)

Proof.

See [

13

].

Theorem

1313

.

1

and

2

are different relations.

Proof.

Consider

a

is an arbitrary non-empty filter. Then

a

1

F

(Base(

a

))

but not

a

2

F

(Base(

a

))

.

Proposition

1314

.

If

a

2

b

where

a

is an ultrafilter then

b

is also an ultra-

filter.

Proof.

b

=

FCD

f

a

for some

f

: Base(

a

)

Base(

b

). So

b

is an ultrafilter

since

f

is monovalued.

Corollary

1315

.

If

a

1

b

where

a

is an ultrafilter then

b

is also an ultrafilter

or

F

(Base(

a

))

.

Proof.

b

v

FCD

f

a

for some

f

: Base(

a

)

Base(

b

). Therefore

b

0

=

FCD

f

a

is an ultrafilter. From this our statement follows.

Proposition

1316

.

Principal filters, generated by sets of the same cardinality,

are isomorphic.

Proof.

Let

A

and

B

be sets of the same cardinality. Then there are a bijection

f

from

A

to

B

. We have

h

f

i

A

=

B

and thus

A

and

B

are isomorphic.

Proposition

1317

.

If a filter is isomorphic to a principal filter, then it is also

a principal filter induced by a set with the same cardinality.

Proof.

Let

A

be a principal filter and

B

is a filter isomorphic to

A

. Then

there are sets

X

A

and

Y

B

such that there are a bijection

f

:

X

Y

such

that

h

f

i

A

=

B

.

So min

B

exists and min

B

=

h

f

i

min

A

and thus

B

is a principal filter (of the

same cardinality as

A

).