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5.39. EQUIVALENT FILTERS AND REBASE OF FILTERS

133

Obvious

745

.

The filtrator of unfixed filters is down-aligned.

Proposition

746

.

The filtrator of unfixed filters is

1

. filtered;

2

. with join-closed core.

Proof.

Theorem

531

.

Proposition

747

.

The filtrator of unfixed filters is with binarily meet-closed

core.

Proof.

Corollary

533

.

Proposition

748

.

The filtrator of unfixed filters is with separable core.

Proof.

Theorem

534

.

Proposition

749

.

Cor

X

and Cor

0

X

are defined for every unfixed filter

X

and

Cor

X

= Cor

0

X

, provided that every

DA

is a complete lattice.

Proof.

Cor

X

and Cor

0

X

exists because of the above isomorphism.

Cor

0

X

= Cor

X

by theorem

542

.

Obvious

750

.

Cor

X

= Cor

0

X

=

T

X

for every filter

X ∈

F

(small sets).

Proposition

751

.

atoms

d

S

=

T

h

atoms

i

S

whenever

d

S

is defined.

Proof.

Theorem

108

.

Proposition

752

.

atoms(

A t B

) = atoms

A ∪

atoms

B

for unfixed filters

A

,

B

,

whenever

Z

is a distributive lattice which is an ideal base.

Proof.

Proposition

554

.

Proposition

753

.

X

is a free star for every unfixed filter

X

, whenever

Z

is

a distributive lattice which is an ideal base which has a least element.

Proof.

Theorem

563

.

Proposition

754

.

The poset of unfixed filters is an atomistic lattice if ev-

ery

DA

(for

A

A

) is an atomistic lattice.

Proof.

Easily follows from

735

by isomorphism.

Proposition

755

.

The poset of unfixed filters is a strongly separable lattice

if every

DA

(for

A

A

) is an atomistic lattice.

Proof.

Theorem

231

.

Proposition

756

.

Cor

X

=

d

(

Z

atoms

unfixed filters

) for every unfixed filter

X

if every

DA

(for

A

A

) is an atomistic lattice.

Proof.

Theorem

596

.

Proposition

757

.

Cor(

A u B

) = Cor

A u

Cor

B

for every unfixed filters

A

,

B

,

provided every

DA

(for

A

A

) is a complete lattice.

Proof.

Theorem

598

.

Proposition

758

.

Cor

d

A

S

=

d

Z

h

Cor

i

S

for the filtrator of unfixed filters

for every nonempty set

S

of unfixed filters, provided every

DA

(for

A

A

) is a

complete lattice.

Proof.

Theorem

599

.