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7.12. COMPLETE FUNCOIDS

172

consequently

l

g

(

X, Y

)

X

T

A, Y

up

h

f

i

X

X

v h

f

i

X

that is

l

g

(

X, Y

)

X

T

A, Y

up

h

f

i

X

v

f

and finally

f

=

l

g

(

X, Y

)

X

T

A, Y

up

h

f

i

X

.

Corollary

925

.

Filtrators of funcoids are filtered.

Theorem

926

.

1

.

g

is metacomplete if

g

is a complete funcoid.

2

.

g

is co-metacomplete if

g

is a co-complete funcoid.

Proof.

1

Let

R

be a set of funcoids from a set

A

to a set

B

and

g

be a funcoid from

B

to some

C

. Then

D

g

l

R

E

X

=

h

g

i

D

l

R

E

X

=

h

g

i

l

f

R

h

f

i

X

=

l

f

R

h

g

ih

f

i

X

=

l

f

R

h

g

f

i

X

=

*

l

f

R

(

g

f

)

+

X

=

D

l

h

g

◦i

R

E

X

for every typed set

X

T

A

. So

g

d

R

=

d

h

g

◦i

R

.

2

By duality.

Conjecture

927

.

g

is complete if

g

is a metacomplete funcoid.

I will denote

ComplFCD

and

CoComplFCD

the sets of small complete and co-

complete funcoids correspondingly.

ComplFCD

(

A, B

) are complete funcoids from

A

to

B

and likewise with

CoComplFCD

(

A, B

).

Obvious

928

.

ComplFCD

and

CoComplFCD

are closed regarding composition

of funcoids.

Proposition

929

.

ComplFCD

and

CoComplFCD

(with induced order) are com-

plete lattices.

Proof.

It follows from theorem

922

.

Theorem

930

.

Atoms of the lattice

ComplFCD

(

A, B

) are exactly funcoidal

products of the form

A

{

α

} ×

FCD

b

where

α

A

and

b

is an ultrafilter on

B

.