background image

Theorem 14.6.

For a lter

a

we have

a

RLD

a

v

id

RLD

(

Base

(

a

))

only in the case if

a

= 0

F

(

Base

(

a

))

or

a

is a trivial ultralter.

Proof.

If

a

RLD

a

v

id

RLD

(

Base

(

a

))

then there exists

m

2

GR

(

a

RLD

a

)

such that

m

id

Base

(

a

)

.

Consequently there exist

A; B

2

GR

a

such that

A

B

id

Base

(

a

)

what is possible only in the case

when

"

Base

(

a

)

A

=

"

Base

(

a

)

B

=

a

is trivial a ultralter or the least lter.

Corollary 14.7.

Reloidal product of a non-trivial atomic lter with itself is non-atomic.

Proof.

Obviously

(

a

RLD

a

)

u

id

RLD

(

Base

(

a

))

=

/ 0

F

(

Base

(

a

))

and

(

a

RLD

a

)

u

id

RLD

(

Base

(

a

))

@

a

RLD

a

.

Example 14.8.

There exist two atomic reloids whose composition is non-atomic and non-empty.

Proof.

Let

a

be a non-trivial ultralter on

N

and

x

2

N

. Then

(

a

RLD

"

N

f

x

g

)

(

"

N

f

x

RLD

a

) =

l

"

RLD

(

N

;

N

)

((

A

 f

x

g

)

(

f

x

A

))

j

A

2

a

 

=

l

"

RLD

(

N

;

N

)

(

A

A

)

j

A

2

a

 

=

a

RLD

a

is non-atomic despite of

a

RLD

"

N

f

x

g

and

"

N

f

x

RLD

a

are atomic.

Example 14.9.

There exists non-monovalued atomic reloid.

Proof.

From the previous example it follows that the atomic reloid

"

N

f

x

RLD

a

is not mono-

valued.

Example 14.10.

Non-convex reloids exist.

Proof.

Let

a

be a non-trivial ultralter. Then id

a

RLD

is non-convex. This follows from the fact

that only reloidal products which are below id

RLD

(

Base

(

a

))

are reloidal products of ultralters and

id

a

RLD

is not their join.

Example 14.11.

(

RLD

)

in

f

=

/ (

RLD

)

out

f

for a funcoid

f

.

Proof.

Let

f

=

id

FCD

(

N

)

. Then

(

RLD

)

in

f

=

a

RLD

a

j

a

2

atoms

F

(

N

)

 

and

(

RLD

)

out

f

=

id

RLD

(

N

)

. But as have shown above

a

RLD

a

v

id

RLD

(

N

)

for non-trivial ultralter

a

, and so

(

RLD

)

in

f

v

(

RLD

)

out

f

.

Proposition 14.12.

id

FCD

(

U

)

u "

FCD

(

U

;

U

)

((

U

U

)

n

id

U

) =

id

(

U

)

FCD

=

/ 0

FCD

(

U

;

U

)

for every innite set

U

.

Proof.

Note that

id

(

U

)

FCD

X

=

X u

(

U

)

for every lter

X

on

U

.

Let

f

=

id

FCD

(

U

)

,

g

=

"

FCD

(

U

;

U

)

((

U

U

)

n

id

U

)

.

Let

x

be a non-trivial ultralter on

U

. If

X

2

x

then card

X

>

2

(In fact,

X

is innite but we

don't need this.) and consequently

h

g

i

X

= 1

F

(

U

)

. Thus

h

g

i

x

= 1

F

(

U

)

. Consequently

h

f

u

g

i

x

=

h

f

i

x

u h

g

i

x

=

x

u

1

F

(

U

)

=

x:

Also

id

(

U

)

FCD

x

=

x

u

(

U

) =

x

.

Let now

x

be a trivial ultralter. Then

h

f

i

x

=

x

and

h

g

i

x

= 1

F

(

U

)

n

x

. So

h

f

u

g

i

x

=

h

f

i

x

u h

g

i

x

=

x

u

¡

1

F

(

U

)

n

x

= 0

F

(

U

)

:

Also

id

(

U

)

FCD

x

=

x

u

(

U

) = 0

F

(

U

)

.

So

h

f

u

g

i

x

=

id

(

U

)

FCD

x

for every ultralter

x

on

U

. Thus

f

u

g

=

id

(

U

)

FCD

.

Example 14.13.

There exist binary relations

f

and

g

such that

"

FCD

(

A

;

B

)

f

u "

FCD

(

A

;

B

)

g

=

/

"

FCD

(

A

;

B

)

(

f

\

g

)

for some sets

A

,

B

such that

f ; g

A

B

.

172

Counter-examples about funcoids and reloids