(used the lemma and theorem

4.43

).

Corollary 4.61.

If

(

A

;

Z

)

is a ltered up-aligned complete lattice ltrator with co-separable core

which is a complete boolean lattice, then

a

+

2

Z

for every

a

2

A

.

Theorem 4.62.

If

(

A

;

Z

)

is a ltered complete lattice ltrator with down-aligned, nitely meet-

closed, separable core which is a complete boolean lattice, then

a

=

Cor

a

=

Cor

0

a

for every

a

2

A

.

Proof.

Our ltrator is with join-closed core (theorem

4.25

).

a

=

F

A

f

c

2

A

j

c

u

A

a

= 0

A

g

. But

c

u

A

a

= 0

A

) 9

C

2

up

c

:

C

u

A

a

= 0

A

. So

a

=

G

A

f

C

2

Z

j

C

u

A

a

= 0

A

g

=

G

A

f

C

2

Z

j

a

v

C

g

=

G

A

f

C

j

C

2

Z

; a

v

C

g

=

G

A

f

C

j

C

2

up

a

g

=

G

Z

f

C

j

C

2

up

a

g

=

l

Z

f

C

j

C

2

up

a

g

=

l

Z

up

a

=

Cor

a

(used theorem

4.43

).

Cor

a

=

Cor

0

a

by theorem

4.34

.

Corollary 4.63.

If

(

A

;

Z

)

is a ltered down-aligned and up-aligned complete lattice ltrator with

nitely meet-closed, separable and co-separable core which is a complete boolean lattice, then

a

=

a

+

for every

a

2

A

.

Proof.

Comparing two last theorems.

Theorem 4.64.

If

(

A

;

Z

)

is a complete lattice ltrator with join-closed separable core which is a

complete lattice, then

a

2

Z

for every

a

2

A

.

Proof.

f

c

2

A

j

c

u

A

a

= 0

A

g  f

A

2

Z

j

A

u

A

a

= 0

A

g

; consequently

a

w

F

A

f

A

2

Z

j

A

u

A

a

= 0

A

g

.

But if

c

2 f

c

2

A

j

c

u

A

a

= 0

A

g

then there exists

A

2

Z

such that

A

w

c

and

A

u

A

a

= 0

A

that is

A

2 f

A

2

Z

j

A

u

A

a

= 0

A

g

. Consequently

a

v

F

A

f

A

2

Z

j

A

u

A

a

= 0

A

g

.

We have

a

=

F

A

f

A

2

Z

j

A

u

A

a

= 0

A

g

=

F

Z

f

A

2

Z

j

A

u

A

a

= 0

A

g 2

Z

.

Theorem 4.65.

If

(

A

;

Z

)

is an up-aligned ltered complete lattice ltrator with co-separable core

which is a complete boolean lattice, then

a

+

is dual pseudocomplement of

a

, that is

a

+

=

min

f

c

2

A

j

c

t

A

a

= 1

A

g

for every

a

2

A

.

Proof.

Our ltrator is with join-closed core (theorem

4.25

). It's enough to prove that

a

+

t

A

a

= 1

A

.

But

a

+

t

A

a

=

Cor

a

t

A

a

w

Cor

a

t

A

Cor

a

=

Cor

a

t

Z

Cor

a

= 1

A

(used the theorem

4.29

and the

fact that our ltrator is ltered).

Denition 4.66.

The

edge part

of an element

a

2

A

is Edg

a

=

a

n

Cor

a

, the

dual edge part

is

Edg

0

a

=

a

n

Cor

0

a

.

4.2 Filtrators

59