* theorem 40 in [15].

Thus Compl

f

is principal.

Theorem 60

Compl CoCompl

f

= CoCompl Compl

f

= Cor

f

for every reloid

f

.

Proof

We will prove only Compl CoCompl

f

= Cor

f

. The rest follows from

symmetry.

From the lemma Compl CoCompl

f

is principal. It is obvious Compl CoCompl

f

f

. So to finish the proof we need to show only that for every principal reloid

F

f

we have

F

Compl CoCompl

f

.

Really, obviously

F

CoCompl

f

and thus

F

= Compl

F

Compl CoCompl

f

.

Question 28.

Is Compl

RLD

(

A

;

B

) a distributive lattice? Is Compl

RLD

(

A

;

B

)

a co-brouwerian lattice?

Conjecture 12

Let

A

,

B

,

C

are small sets. If

f

RLD

(

B

;

C

)

is a complete

reloid and

R

P

RLD

(

A

;

B

)

then

f

[

R

=

[

h

f

◦i

R.

This conjecture can be weakened:

Conjecture 13

Let

A

,

B

,

C

are small sets. If

f

RLD

(

B

;

C

)

is a principal

reloid and

R

P

RLD

(

A

;

B

)

then

f

[

R

=

[

h

f

◦i

R.

Conjecture 14

Compl

f

=

f

\

(Ω

Src

f

×

RLD

1

F

(Dst

f

)

)

for every reloid

f

.

5

Relationships between funcoids and reloids

5.1

Funcoid induced by a reloid

Every reloid

f

induces a funcoid (

FCD

)

f

FCD

(Src

f

; Dst

f

) by the following

formulas (for every

X ∈

F

(Src

f

),

Y ∈

F

(Dst

f

)):

X

[(

FCD

)

f

]

Y ⇔ ∀

F

up

f

:

X

FCD

(Src

f

;Dst

f

)

F

Y

;

h

(

FCD

)

f

i X

=

FCD

(Src

f

;Dst

f

)

F

X

|

F

up

f

.

We should prove that (

FCD

)

f

is really a funcoid.

Proof

We need to prove that

X

[(

FCD

)

f

]

Y ⇔ Y ∩ h

(

FCD

)

f

i X 6

= 0

F

(Dst

f

)

⇔ X ∩

(

FCD

)

f

1

Y 6

= 0

F

(Dst

f

)

.

51