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15. COUNTER-EXAMPLES ABOUT FUNCOIDS AND RELOIDS

253

Proof.

Take

R

+

=

x

R

x>

0

 

,

A

= ∆,

T

=

n

↑{

x

}

x

R

+

o

where

=

R

.

d

T

=

R

+

;

A ×

RLD

d

T

= ∆

×

RLD

R

+

.

d

B∈

T

(

A ×

RLD

B

) =

d

x

R

+

(∆

×

RLD

↑ {

x

}

).

We’ll prove that

d

x

R

+

(∆

×

RLD

↑ {

x

}

)

6

= ∆

×

RLD

R

+

.

Consider

K

=

S

x

R

+

(

{

x

]

1

/x

; 1

/x

[).

K

up(∆

×

RLD

↑ {

x

}

) and thus

K

up

d

x

R

+

(∆

×

RLD

↑ {

x

}

) . But

K /

up(∆

×

RLD

R

+

).

Theorem

1338

.

For a filter

a

we have

a

×

RLD

a

v

1

RLD

Base(

a

)

only in the case if

a

=

F

(Base(

a

))

or

a

is a trivial ultrafilter.

Proof.

If

a

×

RLD

a

v

1

RLD

Base(

a

)

then there exists

m

up(

a

×

RLD

a

) such that

m

v

1

Rel

Base(

a

)

. Consequently there exist

A, B

up

a

such that

A

×

B

v

1

Rel

Base(

a

)

what is possible only in the case when

A

=

B

=

a

is trivial a ultrafilter or the

least filter.

Corollary

1339

.

Reloidal product of a non-trivial atomic filter with itself is

non-atomic.

Proof.

Obviously (

a

×

RLD

a

)

u

1

RLD

Base(

a

)

6

=

RLD

and (

a

×

RLD

a

)

u

1

RLD

Base(

a

)

@

a

×

RLD

a

.

Example

1340

.

There exist two atomic reloids whose composition is non-

atomic and non-empty.

Proof.

Let

a

be a non-trivial ultrafilter on

N

and

x

N

. Then

(

a

×

RLD

N

{

x

}

)

(

N

{

x

} ×

RLD

a

) =

RLD

(

N

,

N

)

l

A

a

((

A

× {

x

}

)

(

{

x

} ×

A

) =

RLD

(

N

,

N

)

l

A

a

(

A

×

A

) =

a

×

RLD

a

is non-atomic despite of

a

×

RLD

N

{

x

}

and

N

{

x

} ×

RLD

a

are atomic.

Example

1341

.

There exists non-monovalued atomic reloid.

Proof.

From the previous example it follows that the atomic reloid

N

{

x

} ×

RLD

a

is not monovalued.

Example

1342

.

Non-convex reloids exist.

Proof.

Let

a

be a non-trivial ultrafilter. Then id

RLD

a

is non-convex. This

follows from the fact that only reloidal products which are below 1

RLD

Base(

a

)

are reloidal

products of ultrafilters and id

RLD

a

is not their join.

Example

1343

.

There exists (atomic) composable funcoids

f

and

g

such that

H

up(

g

f

)

;

F

up

f, G

up

g

:

H

w

G

F.

Proof.

Let

a

be a nontrivial ultrafilter and

p

be an arbitrary point,

f

=

a

×

FCD

{

p

}

,

g

=

{

p

} ×

FCD

a

. Then

g

f

=

a

×

FCD

a

. Take

H

= 1. Let

F

up

f

and

G

up

g

. We have

F

up(

A

0

×

FCD

{

p

}

),

G

up(

{

p

} ×

FCD

A

1

) where

A

0

, A

1

up

a

(take

A

0

=

h

F

i

@

{

p

}

and similarly for

A

1

). Thus

G

F

w

A

0

×

A

1

and so

H /

up(

G

F

).

Example

1344

.

(

RLD

)

in

f

6

= (

RLD

)

out

f

for a funcoid

f

.