14. COUNTER-EXAMPLES ABOUT FUNCOIDS AND RELOIDS

194

Proof.

Take

R

+

=

x

R

x>

0

,

A

= ∆,

T

=

n

↑{

x

}

x

R

+

o

where

=

R

.

F

T

=

R

+

;

A ×

RLD

F

T

= ∆

×

RLD

R

+

.

F

n

RLD

B

B∈

T

o

=

F

n

×

RLD

↑{

x

}

x

R

+

o

.

We’ll prove that

F

n

×

RLD

↑{

x

}

x

R

+

o

6

= ∆

×

RLD

R

+

.

Consider

K

=

S

n

{

x

(

1

/x

;1

/x

)

x

R

+

o

.

K

GR(∆

×

RLD

↑ {

x

}

) and thus

K

GR

F

n

×

RLD

↑{

x

}

x

R

+

o

. But

K /

GR(∆

×

RLD

R

+

).

Theorem

1029

.

For a filter

a

we have

a

×

RLD

a

v

id

RLD

(Base(

a

))

only in the

case if

a

=

F

(Base(

a

))

or

a

is a trivial ultrafilter.

Proof.

If

a

×

RLD

a

v

id

RLD

(Base(

a

))

then there exists

m

GR(

a

×

RLD

a

) such

that

m

id

Base(

a

)

. Consequently there exist

A, B

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 ultrafilter or the least filter.

Corollary

1030

.

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

non-atomic.

Proof.

Obviously (

a

×

RLD

a

)

u

id

RLD

(Base(

a

))

6

=

F

(Base(

a

))

and (

a

×

RLD

a

)

u

id

RLD

(Base(

a

))

@

a

×

RLD

a

.

Example

1031

.

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

) =

l

RLD

(

N

;

N

)

((

A

× {

x

}

)

(

{

x

} ×

A

)

A

a

=

l

RLD

(

N

;

N

)

(

A

×

A

)

A

a

=

a

×

RLD

a

is non-atomic despite of

a

×

RLD

N

{

x

}

and

N

{

x

} ×

RLD

a

are atomic.

Example

1032

.

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

1033

.

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 id

RLD

(Base(

a

))

are

reloidal products of ultrafilters and id

RLD

a

is not their join.

Example

1034

.

(

RLD

)

in

f

6

= (

RLD

)

out

f

for a funcoid

f

.

Proof.

Let

f

= id

FCD

(

N

)

. Then (

RLD

)

in

f

=

F

n

a

×

RLD

a

a

atoms

F

(

N

)

o

and (

RLD

)

out

f

=

id

RLD

(

N

)

. But we have shown above

a

×

RLD

a

6v

id

RLD

(

N

)

for non-trivial ultrafilter

a

,

and so (

RLD

)

in

f

6v

(

RLD

)

out

f

.

Proposition

1035

.

id

FCD

(

U

)

u ↑

FCD

(

U

;

U

)

((

U

×

U

)

\

id

U

) = id

FCD

Ω(

U

)

6

=

FCD

(

U

;

U

)

for every infinite set

U

.