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Example 4

Non-convex reloids exist.

Proof

Let

a

is a non-trivial atomic f.o. Then id

RLD

a

is non-convex. This follows

from the fact that only direct products which are below id

RLD(Base(

a

))

are direct

products of atomic f.o. and id

RLD

a

is not their join.

Example 5

(

RLD

)

in

f

6

= (

RLD

)

out

f

for a funcoid

f

.

Proof

Let

f

=

I

FCD

(

N

)

. Then (

RLD

)

in

f

=

a

×

RLD

a

|

a

atoms 1

F

(

N

)

 

and (

RLD

)

out

f

=

I

RLD

(

N

)

. But as we have shown above

a

×

RLD

a

*

I

RLD

(

N

)

for

non-trivial f.o.

a

, and so (

RLD

)

in

f

*

(

RLD

)

out

f

.

Proposition 51

I

FCD

(

N

)

∩ ↑

FCD

(

N

;

N

)

((

N

×

N

)

\

I

N

) =

I

FCD

Ω(

N

)

6

= 0

FCD

(

N

;

N

)

.

Proof

Note that

D

I

FCD

Ω(

N

)

E

X

=

X ∩

Ω (

N

).

Let

f

=

I

FCD

(

N

)

,

g

=

FCD

(

N

;

N

)

((

N

×

N

)

\

I

N

).

Let

x

is a non-trivial atomic f.o. If

X

up

x

then card

X

>

2 (In fact,

X

is infinite but we don’t need this.) and consequently

h

g

i

X

= 1

F

(

N

)

. Thus

h

g

i

x

= 1

F

(

N

)

. Consequently

h

f

g

i

x

=

h

f

i

x

∩ h

g

i

x

=

x

1

F

(

N

)

=

x.

Also

D

I

FCD

Ω(

N

)

E

x

=

x

Ω (

N

) =

x

.

Let now

x

is a trivial f.o. Then

h

f

i

x

=

x

and

h

g

i

x

= 1

F

(

N

)

\

x

. So

h

f

g

i

x

=

h

f

i

x

∩ h

g

i

x

=

x

1

F

(

N

)

\

x

= 0

F

(

N

)

.

Also

D

I

FCD

Ω(

N

)

E

x

=

x

Ω (

N

) = 0

F

(

N

)

.

So

h

f

g

i

x

=

D

I

FCD

Ω(

N

)

E

x

for every atomic f.o.

x

. Thus

f

g

=

I

FCD

Ω(

N

)

.

Example 6

There exist binary relations

f

and

g

such that

FCD

(

A

;

B

)

f

∩ ↑

FCD

(

A

;

B

)

g

6

=

FCD

(

A

;

B

)

(

f

g

) for some sets

A

,

B

such that

f, g

A

×

B

.

Proof

From the proposition above.

Example 7

There exists a principal funcoid which is not a complemented ele-

ment of the lattice of funcoids.

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