4.4. FILTERS ON A SET

80

Proposition

486

.

Cor

a

=

n

p

U

↑{

p

}v

a

o

and

T

a

=

n

p

U

↑{

p

}v

a

o

for every filter

a

on a set.

Proof.

By propositions

415

and

448

.

Proposition

487

.

For every filter

a

on a set

a

=

a

+

=

Cora

=

Cor

0

a

.

Proof.

By propositions

416

and

448

.

Corollary

488

.

For every filter

a

on a set

a

=

a

+

P

.

Proposition

489

.

If

a

is a filter on a set, then

a

+

is dual pseudocomplement

of

a

, that is

a

+

= min

c

F

c

t

F

a

=

>

F

.

Proof.

By proposition

418

.

Proposition

490

.

If

a

,

b

are filters on a set, then

1

.

T

(

a

u

F

b

) =

T

a

T

b

;

2

.

T

(

a

t

F

b

) =

T

a

T

b

.

Proof.

By propositions

420

and

423

.

Proposition

491

.

T

d

F

S

=

T

h

T

i

S

.

Proof.

By proposition

421

.

Proposition

492

.

If

a

,

b

are filters on a set, then

1

. (

a

u

F

b

)

=

a

t

P

b

;

2

. (

a

t

F

b

)

=

a

u

P

b

.

Proof.

By propositions

424

and

425

.

Proposition

493

.

For every

X, Y

P

U

and filter

F

on

U

we have:

X

∼↑

Y

⇔ ∃

A

∈ A

:

X

A

=

Y

A.

Proof.

By theorem

429

.

Proposition

494

.

Let

F

be the set of filters on a set

U

and

A ∈

F

. Consider

the function

γ

:

Z

(

D

A

)

(

P

U

)

/

defined by the formula (for every

p

Z

(

D

A

))

γp

=

X

Z

X

u

F

A

=

p

.

Then:

1

.

γ

is a lattice isomorphism.

2

.

Q

q

:

γ

1

q

=

Q

u

F

A

for every

q

(

P

U

)

/

.

Proof.

By theorem

432

.

Proposition

495

.

(

P

U

)

/

is a boolean lattice.

Proof.

By corollary

433

.

Proposition

496

.

For a lattice

F

of filters on a set and

a, b

F

the following

expressions are always equal:

1

.

a

\

b

=

d

n

z

F

a

v

b

t

z

o

(quasidifference of

a

and

b

);

2

.

a

#

b

=

n

z

F

z

v

a

z

u

b

=

o

(second quasidifference of

a

and

b

);

3

.

F

(atoms

a

\

atoms

b

).

Proof.

Theorem

434

.