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17. GENERALIZED COFINITE FILTERS

275

1

2

Obvious.

2

3

Because complete atomic boolean lattice is isomorphic to a powerset.

3

4

Theorems

1443

and

154

.

Proposition

1446

.

1

.

h

FCD

i{

x

}

= Ω

U

;

2

.

h

FCD

i

p

=

>

for every nontrivial atomic filter

p

.

Proof.

h

FCD

i{

x

}

=

d

A

y

U

(

U

\ {

y

}

) = Ω

U

;

h

FCD

i

p

=

d

A

y

U

>

=

>

.

Proposition

1447

.

(

FCD

)Ω

RLD

= Ω

FCD

.

Proof.

(

FCD

)Ω

RLD

=

d

FCD

up Ω

RLD

= Ω

FCD

.

Proposition

1448

.

(

RLD

)

out

FCD

= Ω

RLD

.

Proof.

(

RLD

)

out

FCD

=

d

RLD

up Ω

FCD

=

d

RLD

up Ω

RLD

= Ω

RLD

.

Proposition

1449

.

(

RLD

)

in

FCD

= Ω

RLD

.

Proof.

(

RLD

)

in

FCD

=

l

a

×

RLD

b

a

atoms

F

, b

atoms

F

, a

×

FCD

b

v

FCD

=

l

a

×

RLD

b

a

atoms

F

, b

atoms

F

,

not

a

and

b

both trivial

=

l

d

atoms(

a

×

RLD

b

)

a

atoms

F

, b

atoms

F

,

not

a

and

b

both trivial

=

l

[

atoms(

a

×

RLD

b

)

a

atoms

F

, b

atoms

F

,

not

a

and

b

both trivial

=

l

(nontrivial atomic reloids under

A

×

B

) = Ω

RLD

.