Example 1

There exist a funcoid

f

and a set

S

of funcoids such that

f

S

S

6

=

S

h

f

∩i

S

.

Proof

Let

f

= ∆

×

FCD

F

(

R

)

{

0

}

and

S

=

FCD

(

R

;

R

)

((

ε

; +

)

× {

0

}

)

|

ε >

0

.

Then

f

S

S

= (∆

×

FCD

F

(

R

)

{

0

}

)

∩ ↑

FCD

(

R

;

R

)

((0; +

)

× {

0

}

) = (∆

∩ ↑

F

(

R

)

(0; +

))

×

FCD

F

(

R

)

{

0

} 6

= 0

FCD

(

R

;

R

)

while

S

h

f

∩i

S

=

0

FCD

(

R

;

R

)

= 0

FCD

(

R

;

R

)

.

Example 2

There exist a set

R

of funcoids and a funcoid

f

such that

f

S

R

6

=

S

h

f

◦i

R

.

Proof

Let

f

= ∆

×

FCD

{

0

}

,

R

=

{

0

} ×

FCD

(

ε

; +

)

|

ε

R

.

We have

S

R

=

{

0

} ×

FCD

(0; +

);

f

S

R

=

FCD

(

R

;

R

)

(

{

0

} × {

0

}

)

6

=

0

FCD

(

R

;

R

)

and

S

h

f

◦i

R

=

0

FCD

(

R

;

R

)

= 0

FCD

(

R

;

R

)

.

Example 3

There exist a set

R

of funcoids and f.o.

X

and

Y

such that

1.

X

[

S

R

]

Y ∧

f

R

:

X

[

f

]

Y

;

2.

h

S

R

i X ⊃

S

{h

f

i X

|

f

R

}

.

Proof

1. Let

X

= ∆ and

Y

= 1

F

(

R

)

. Let

R

=

FCD

(

R

;

R

)

((

ε

; +

)

×

R

)

|

ε

R

, ε >

0

.

Then

S

R

=

FCD

(

R

;

R

)

((0; +

)

×

R

). So

X

[

S

R

]

Y

and

f

R

:

¬

(

X

[

f

]

Y

).

2. With the same

X

and

R

we have

h

S

R

i X

=

R

and

h

f

i X

= 0

F

(

R

)

for every

f

R

, thus

S

{h

f

i X

|

f

R

}

= 0

F

(

R

)

.

Theorem 78

For a f.o.

a

we have

a

×

RLD

a

I

RLD

(Base(

a

))

only in the case if

a

= 0

F

(Base(

a

))

or

a

is a trivial atomic f.o. (that is corresponds to an one-element

set).

Proof

If

a

×

RLD

a

I

RLD

(Base(

a

))

then exists

m

up(

a

×

RLD

a

) such that

m

I

Base(

a

)

. Consequently exist

A, B

up

a

such that

A

×

B

I

Base(

a

)

what

is possible only in the case when

Base(

a

)

A

=

Base(

a

)

B

=

a

and

A

=

B

is an

one-element set or empty set.

Corollary 22

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

is non-atomic.

Proof

Obviously (

a

×

RLD

a

)

I

RLD

(Base(

a

))

6

= 0

F

(Base(

a

))

and (

a

×

RLD

a

)

I

RLD

(Base(

a

))

a

×

RLD

a

.

71