Theorem 36

h

CoCompl

f

i

X

= Cor

h

f

i

X

for every funcoid

f

and set

X

P

(Src

f

)

.

Proof

CoCompl

f

f

thus

h

CoCompl

f

i

X

⊆ h

f

i

X

, but

h

CoCompl

f

i

X

is a principal f.o. thus

h

CoCompl

f

i

X

Cor

h

f

i

X

.

Let

αX

= Cor

h

f

i

X

. Then

α

= 0

F

(Dst

f

)

and

α

(

X

Y

) = Cor

h

f

i

(

X

Y

) = Cor(

h

f

i

X

∪h

f

i

Y

) = Cor

h

f

i

X

Cor

h

f

i

Y

=

αX

αY.

(used the theorem 65 from [15]). Thus

α

can be continued till

h

g

i

for some

funcoid

g

. This funcoid is co-complete.

Evidently

g

is the greatest co-complete element of

FCD

(Src

f

; Dst

f

) which

is lower than

f

.

Thus

g

= CoCompl

f

and so Cor

h

f

i

X

=

αX

=

h

g

i

X

=

h

CoCompl

f

i

X

.

Theorem 37

Compl

FCD

(

A

;

B

)

is an atomistic lattice.

Proof

Let

f

Compl

FCD

(

A

;

B

).

h

f

i

X

=

h

f

i

{

x

}

|

x

X

=

S

n

f

|

Src

f

{

x

}

{

x

}

|

x

X

o

=

S

n

f

|

Src

f

{

x

}

X

|

x

X

o

, thus

f

=

f

|

Src

f

{

x

}

|

x

X

. It is trivial that every

f

|

Src

f

{

x

}

is a join of atoms

of Compl

FCD

(

A

;

B

).

Theorem 38

A funcoid

f

is complete iff it is a join (on the lattice

FCD

(Src

f

; Dst

f

)

)

of atomic complete funcoids.

Proof

It follows from the theorem 29 and the previous theorem.

Corollary 11

Compl

FCD

(

A

;

B

)

is join-closed.

Theorem 39

Compl(

S

R

) =

S

h

Compl

i

R

for every

R

P

FCD

(

A

;

B

)

(for

every small sets

A

,

B

).

Proof

h

Compl(

S

R

)

i

X

=

h

S

R

i

{

α

}

|

α

X

=

S S

h

f

i

{

α

}

|

f

R

|

α

X

=

S S

h

f

i

{

α

}

|

α

X

|

f

R

=

h

Compl

f

i

X

|

f

R

=

h

S

h

Compl

i

R

i

X

for every set

X

.

Corollary 12

Compl

Conjecture 5

Compl

is not an upper adjoint (in general).

Proposition 25

Compl

f

=

f

|

Src

f

{

α

}

|

α

Src

f

for every funcoid

f

.

35