 14.3. CONSEQUENCES

248

Proposition

1318

.

A filter isomorphic to a non-trivial ultrafilter is a non-

trivial ultrafilter.

Proof.

Let

a

be a non-trivial ultrafilter and

a

be isomorphic to

b

. Then

a

2

b

and thus

b

is an ultrafilter. The filter

b

cannot be trivial because otherwise

a

would

be also trivial.

Theorem

1319

.

For an infinite set

U

there exist 2

2

card

U

equivalence classes of

isomorphic ultrafilters.

Proof.

The number of bijections between any two given subsets of

U

is no

more than (card

U

)

card

U

= 2

card

U

. The number of bijections between all pairs of

subsets of

U

is no more than 2

card

U

·

2

card

U

= 2

card

U

. Therefore each isomorphism

class contains at most 2

card

U

ultrafilters. But there are 2

2

card

U

ultrafilters. So there

are 2

2

card

U

classes.

Remark

1320

.

One of the above mentioned equivalence classes contains trivial

ultrafilters.

Corollary

1321

.

There exist non-isomorphic nontrivial ultrafilters on any

infinite set.

14.3. Consequences

Theorem

1322

.

The graph of reloid

F ×

RLD

A

{

a

}

is isomorphic to the filter

F

for every set

A

and

a

A

.

Proof.

From

1309

.

Theorem

1323

.

If

f

,

g

are reloids,

f

v

g

and

g

is monovalued then

g

|

dom

f

=

f

.

Proof.

It’s simple to show that

f

=

d

n

f

|

a

a

atoms

F

(Src

f

)

o

(use the fact that

k

v

f

|

a

for some

a

atoms

F

(Src

f

)

for every

k

atoms

f

and the fact that

RLD

(Src

f,

Dst

f

) is atomistic).

Suppose that

g

|

dom

f

6

=

f

. Then there exists

a

atoms dom

f

such that

g

|

a

6

=

f

|

a

.

Obviously

g

|

a

w

f

|

a

.

If

g

|

a

A

f

|

a

then

g

|

a

is not atomic (because

f

|

a

6

=

RLD

(Src

f,

Dst

f

)

) what

contradicts to a theorem above. So

g

|

a

=

f

|

a

what is a contradiction and thus

g

|

dom

f

=

f

.

Corollary

1324

.

Every monovalued reloid is a restricted principal monoval-

ued reloid.

Proof.

Let

f

be a monovalued reloid. Then there exists a function

F

GR

f

.

So we have

(

RLD

(Src

f,

Dst

f

)

F

)

|

dom

f

=

f.

Corollary

1325

.

Every monovalued injective reloid is a restricted injective

monovalued principal reloid.

Proof.

Let

f

be a monovalued injective reloid. There exists a function

F

such

that

f

= (

RLD

(Src

f,

Dst

f

)

F

)

|

dom

f

. Also there exists an injection

G

up

f

.