background image

4.2. FILTRATORS

57

Proof.

Our filtrator is with join-closed core (theorem

292

).

a

=

F

A

c

A

c

u

A

a

=

A

 

. But

c

u

A

a

=

A

⇒ ∃

C

up

c

:

C

u

A

a

=

A

. So

a

=

A

G

C

Z

C

u

A

a

=

A

=

A

G

C

Z

a

v

C

=

A

G

C

C

Z

, a

v

C

=

A

G

C

C

up

a

=

Z

G

C

C

up

a

=

Z

l

C

C

up

a

=

Z

l

up

a

=

Cor

a

(used theorem

310

).

Cor

a

= Cor

0

a

by theorem

301

.

Corollary

330

.

If (

A

;

Z

) is a filtered down-aligned and up-aligned complete

lattice filtrator with finitely meet-closed, separable and co-separable core which is

a complete boolean lattice, then

a

=

a

+

for every

a

A

.

Proof.

Comparing two last theorems.

Theorem

331

.

If (

A

;

Z

) is a complete lattice filtrator with join-closed separable

core which is a complete lattice, then

a

Z

for every

a

A

.

Proof.

c

A

c

u

A

a

=

A

 

A

Z

A

u

A

a

=

A

 

; consequently

a

w

F

A

A

Z

A

u

A

a

=

A

 

.

But if

c

c

A

c

u

A

a

=

A

 

then there exists

A

Z

such that

A

w

c

and

A

u

A

a

=

A

that is

A

A

Z

A

u

A

a

=

A

 

. Consequently

a

v

F

A

A

Z

A

u

A

a

=

A

 

.

We have

a

=

F

A

A

Z

A

u

A

a

=

A

 

=

F

Z

A

Z

A

u

A

a

=

A

 

Z

.

Theorem

332

.

If (

A

;

Z

) is an up-aligned filtered complete lattice filtrator with

co-separable core which is a complete boolean lattice, then

a

+

is dual pseudocom-

plement of

a

, that is

a

+

= min

c

A

c

t

A

a

=

>

A

for every

a

A

.

Proof.

Our filtrator is with join-closed core (theorem

292

). It’s enough to

prove that

a

+

t

A

a

=

>

A

. But

a

+

t

A

a

= Cor

a

t

A

a

w

Cor

a

t

A

Cor

a

= Cor

a

t

Z

Cor

a

=

>

A

(used the theorem

296

and the fact that our filtrator is filtered).

Definition

333

.

The

edge part

of an element

a

A

is Edg

a

=

a

\

Cor

a

, the

dual edge part

is Edg

0

a

=

a

\

Cor

0

a

.