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Theorem 7.56.

Atoms of the lattice Compl

RLD

(

A

;

B

)

are exactly reloidal products of the form

"

A

f

RLD

b

where

2

A

and

b

is an ultralter on

B

.

Proof.

First, it's easy to see that

"

A

f

RLD

b

are elements of Compl

RLD

(

A

;

B

)

. Also

0

RLD

(

A

;

B

)

is an element of Compl

RLD

(

A

;

B

)

.

"

A

f

RLD

b

are atoms of Compl

RLD

(

A

;

B

)

because they are atoms of

RLD

(

A

;

B

)

.

It remains to prove that if

f

is an atom of Compl

RLD

(

A

;

B

)

then

f

=

"

A

f

RLD

b

for some

2

A

and an ultralter

b

on

B

.

Suppose

f

is a non-empty complete reloid. Then

"

A

f

RLD

b

v

f

for some

2

A

and an

ultralter

b

on

B

. If

f

is an atom then

f

=

"

A

f

RLD

b

.

Obvious 7.57.

Compl

RLD

(

A

;

B

)

is an atomistic lattice.

Proposition 7.58.

Compl

f

=

f

j

"

Src

f

f

g

j

2

Src

f

 

for every reloid

f

.

Proof.

Let's denote

R

the right part of the equality to be proven.

That

R

is a complete reloid follows from the equality

f

j

"

Src

f

f

g

=

"

Src

f

f

RLD

im

¡

f

j

"

Src

f

f

g

:

The only thing left to prove is that

g

v

R

for every complete reloid

g

such that

g

v

f

.

Really let

g

be a complete reloid such that

g

v

f

. Then

g

=

G

f"

Src

f

f

RLD

G

(

)

j

2

Src

f

g

for some function

G

:

Src

f

!

F

(

Dst

f

)

.

We have

"

Src

f

f

RLD

G

(

) =

g

j

"

Src

f

f

g

v

f

j

"

Src

f

f

g

. Thus

g

v

R

.

Conjecture 7.59.

Compl

f

u

Compl

g

=

Compl

(

f

u

g

)

for every

f ; g

2

RLD

(

A

;

B

)

.

Theorem 7.60.

Compl

F

R

=

F

h

Compl

i

R

for every set

R

2

P

RLD

(

A

;

B

)

for every sets

A

,

B

.

Proof.

Compl

G

R

=

G ¡G

R

j

"

A

f

g

j

2

A

 

=

(proposition

4.194

)

G G

f

f

j

"

A

f

g

j

2

A

g j

f

2

R

 

=

G

h

Compl

i

R:

Lemma 7.61.

Completion of a co-complete reloid is principal.

Proof.

Let

f

be a co-complete reloid. Then there is a function

F

:

Dst

f

!

F

(

Src

f

)

such that

f

=

G

f

F

(

)

RLD

"

Dst

f

f

g j

2

Dst

f

g

:

So

Compl

f

=

G ¡G

f

F

(

)

RLD

"

Dst

f

f

g j

2

Dst

f

g

j

"

Src

f

f

g

j

2

Src

f

 

=

G ¡G

f

F

(

)

RLD

"

Dst

f

f

g j

2

Dst

f

g

u

¡

"

Src

f

f

RLD

1

F

(

Dst

f

)

j

2

Src

f

 

=

(*)

G G 

(

F

(

)

RLD

"

Dst

f

f

g

)

u

¡

"

Src

f

f

RLD

1

F

(

Dst

f

)

j

2

Dst

f

 

j

2

Src

f

 

=

G G

f"

Src

f

f

RLD

"

Dst

f

f

g j

2

Dst

f

g j

2

Src

f ;

"

Src

f

f

g v

F

(

)

 

:

* proposition

4.194

.

Thus Compl

f

is principal.

Theorem 7.62.

Compl CoCompl

f

=

CoCompl Compl

f

=

Cor

f

for every reloid

f

.

7.7 Complete reloids and completion of reloids

129