 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

v

f

. So to nish

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

F

v

f

we have

F

v

Compl CoCompl

f

.

Really, obviously

F

v

CoCompl

f

and thus

F

=

Compl

F

v

Compl CoCompl

f

.

Question 7.63.

Is Compl

RLD

(

A

;

B

)

a distributive lattice? Is Compl

RLD

(

A

;

B

)

a co-brouwerian

lattice?

Conjecture 7.64.

If

f

is a complete reloid, then it is metacomplete.

Conjecture 7.65.

If

f

is a metacomplete reloid, then it is complete.

Conjecture 7.66.

Compl

f

=

f

n

¡

Src

f

RLD

1

F

(

Dst

f

)

for every reloid

f

.

By analogy with similar properties of funcoids described above:

Proposition 7.67.

For composable reloids

f

and

g

it holds

1. Compl

(

g

f

)

w

(

Compl

g

)

(

Compl

f

)

;

2. CoCompl

(

g

f

)

w

(

CoCompl

g

)

(

CoCompl

f

)

.

Proof.

1.

(

Compl

g

)

(

Compl

f

)

v

Compl

((

Compl

g

)

(

Compl

f

))

v

Compl

(

g

f

)

.

2. By duality.

Conjecture 7.68.

For composable reloids

f

and

g

it holds

1. Compl

(

g

f

) = (

Compl

g

)

f

if

f

is a co-complete reloid;

2. CoCompl

(

f

g

) =

f

CoCompl

g

if

f

is a complete reloid;

3. CoCompl

((

Compl

g

)

f

) =

Compl

(

g

(

CoCompl

f

)) = (

Compl

g

)

(

CoCompl

f

)

;

4. Compl

(

g

(

Compl

f

)) =

Compl

(

g

f

)

;

5. CoCompl

((

CoCompl

g

)

f

) =

CoCompl

(

g

f

)

.

130

Reloids