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Proof.

Suppose that

f

and

g

are two different bijective reloids from

A

to

B

. Then

g

1

f

is not

the identity reloid (otherwise

g

1

f

=

I

dom

f

RLD

and so

f

=

g

). But

g

1

f

is a bijective reloid (as a

composition of bijective reloids) from

A

to

A

what is impossible.

4 Rudin-Keisler equivalence and Rudin-Keisler order

Theorem 61.

Atomic filter objects

a

and

b

(with possibly different bases) are isomorphic iff

a

>

b

b

>

a

.

Proof.

Let

a

>

b

b

>

a

. Then there are a monovalued reloids

f

and

g

such that dom

f

=

a

and

im

f

=

b

and dom

g

=

b

and im

g

=

a

. Thus

g

f

is a monovalued morphism from

a

to

a

. By the

above we have

g

f

=

I

a

RLD

so

g

=

f

1

and

f

1

f

=

I

a

RLD

so

f

is monovalued. Thus

f

is an injective

monovalued reloid from

a

to

b

and thus

a

and

b

are isomorphic.

The last theorem cannot be generalized from atomic f.o. to arbitrary f.o., as it’s shown by the

following two examples:

Example 62.

A

>

1

B ∧ B

>

1

A

but

A

is not isomorphic to

B

for some f.o.

A

and

B

.

Proof.

Consider

A

=

R

[0; 1]

and

B

=

T

{↑

R

[0; 1 +

ε

)

|

ε >

0

}

. Then the function

f

=

{

(

x

;

x

/2)

|

x

R

}

witnesses both inequalities

A

>

1

B

and

B

>

1

A

. But these filters cannot be isomorphic

because only one of them is principal.

Lemma 63.

Let

f

0

and

f

1

are

Set

-morphisms. Let

f

(

x

;

y

) = (

f

0

x

;

f

1

y

)

for a function

f

. Then

FCD

(

Dst

f

0

;

Dst

f

1

)

f

(

A ×

RLD

B

) =

h↑

f

0

iA ×

RLD

h↑

f

1

iB

.

Proof.

FCD

(

Dst

f

0

;

Dst

f

1

)

f

(

A ×

RLD

B

) =

FCD

(

Dst

f

0

;

Dst

f

1

)

f

T

{↑

Src

f

0

×

Src

f

1

(

A

×

B

)

|

A

up

A

,

B

up

B}

=

T

{↑

Src

f

0

×

Src

f

1

h

f

i

(

A

×

B

)

|

A

up

A

, B

up

B}

=

T

{↑

Dst

f

0

×

Dst

f

1

(

h

f

0

i

A

×

h

f

1

i

B

)

|

A

up

A

, B

up

B }

=

T

{↑

Dst

f

0

h

f

0

i

A

×

RLD

Dst

f

1

h

f

1

i

B

|

A

up

A

, B

up

B}

=

(theorem 164?? in [3])

=

T

{↑

Dst

f

0

h

f

0

i

A

|

A

up

A} ×

RLD

T

{↑

Dst

f

1

h

f

1

i

B

|

A

up

B}

=

h↑

f

0

iA ×

RLD

h↑

f

1

iB

.

Lemma 64.

If an f.o.

A

is isomorphic to an f.o.

B

then if

X

is a set and

Base

(

A

)

X

∩ A

is an

atomic f.o., then there exists a set

Y

such that

Base

(

A

)

X

∩ A

is an atomic f.o. isomorphic to

Base

(

A

)

Y

∩ B

.

[FIXME: See the book for a corrected proof.]

Proof.

Let

A

is isomorphic to

B

. Then there are sets

A

up

A

,

B

up

B

such that

A ÷

A

is directly

isomorphic to

B ÷

B

. So there are a bijection

f

:

P

A

up

A →

P

B

up

B

such that

B

=

h

f

iA

.

[FIXME: ?? equality is wrong.]

up

Base

(

A

)

X

∩ A

=

up

Base

(

A

)

(

X

A

)

∩ A

= ?? =

h

X

A

∩ i

up

A

=

h

X

∩ i

(

P

A

up

A

)

.

Thus

hh

f

ii

up

Base

(

A

)

X

∩ A

=

hh

f

iih

X

∩ i

(

P

A

up

A

) =

h

f

(

X

)

∩ ihh

f

ii

(

P

A

up

A

) =

h

f

(

X

)

∩ i

(

P

B

up

B

) =

h

f

(

X

)

B

∩ i

up

B

=

h

f

(

X

)

∩ i

up

B

=

up

Base

(

B

)

(

f

(

X

))

∩ B

.

So

h

f

i ↑

Base

(

A

)

X

A

=

T

Base

(

B

)

hh

f

ii

up

Base

(

A

)

X

A

=

T

Base

(

B

)

up

Base

(

B

)

(

f

(

X

))

∩ B

=

Base

(

B

)

(

f

(

X

))

∩ B

.

Finally we have

Base

(

B

)

(

f

(

X

))

∩ B

is isomorphic to

Base

(

A

)

X

∩ A

from the last equality.

Theorem 65.

Let

f

is a monovalued injective reloid. Then

f

is isomorphic to the f.o. dom

f

.

Proof.

Let

f

is a monovalued injective reloid. There exists a bijection

F

up

f

. Consider the

bijective function

p

=

{

(

x

;

Fx

)

|

x

dom

F

}

.

h

p

i

dom

F

=

F

and consequently

h

p

i

dom

f

=

T

RLD

(

Dst

f

;

Src

f

)

h

p

i

dom

K

|

K

up

f

 

=

T

RLD

(

Dst

f

;

Src

f

)

h

p

i

dom

(

K

F

)

|

K

up

f

 

=

T

RLD

(

Dst

f

;

Src

f

)

(

K

F

)

|

K

up

f

 

=

RLD

(

Dst

f

;

Src

f

)

K

|

K

up

f

 

=

f

. Thus

p

witnesses that

f

is isomorphic to the f.o. dom

f

.

Corollary 66.

A monovalued injective reloid with atomic domain is atomic.

Rudin-Keisler equivalence and Rudin-Keisler order

11