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4.2. FILTRATORS

50

Definition

284

.

I call a filtrator

with meet-closed core

such a filtrator (

A

;

Z

)

that

d

Z

S

=

d

A

S

whenever

d

Z

S

exists for

S

P

Z

.

Definition

285

.

I call a filtrator

with finitely join-closed core

such a filtrator

(

A

;

Z

) that

a

t

Z

b

=

a

t

A

b

whenever

a

t

Z

b

exists for

a, b

Z

.

Definition

286

.

I call a filtrator

with finitely meet-closed core

such a filtrator

(

A

;

Z

) that

a

u

Z

b

=

a

u

A

b

whenever

a

u

Z

b

exists for

a, b

Z

.

Definition

287

.

Filtered filtrator

is a filtrator (

A

;

Z

) such that

a

A

:

a

=

d

A

up

a

.

Definition

288

.

Prefiltered filtrator

is a filtrator (

A

;

Z

) such that “up” is in-

jective.

Definition

289

.

Semifiltered filtrator

is a filtrator (

A

;

Z

) such that

a, b

A

: (up

a

up

b

a

v

b

)

.

Obvious

290

.

Every filtered filtrator is semifiltered.

Every semifiltered filtrator is prefiltered.

Obvious

291

.

“up” is a straight map from

A

to the dual of the poset

P

Z

if

(

A

;

Z

) is a semifiltered filtrator.

Theorem

292

.

Each semifiltered filtrator is a filtrator with join-closed core.

Proof.

Let (

A

;

Z

) be a semifiltered filtrator. Let

S

P

Z

and

F

Z

S

be defined.

We need to prove

F

A

S

=

F

Z

S

. That

F

Z

S

is an upper bound for S is obvious.

Let

a

A

be an upper bound for

S

. It’s enough to prove that

F

Z

S

v

a

. Really,

c

up

a

c

w

a

⇒ ∀

x

S

:

c

w

x

c

w

Z

G

S

c

up

Z

G

S

;

so up

a

up

F

Z

S

and thus

a

w

F

Z

S

because it is semifiltered.

4.2.1. Core Part.

Definition

293

.

The

core part

of an element

a

A

is Cor

a

=

d

Z

up

a

.

Definition

294

.

The

dual core part

of an element

a

A

is Cor

0

a

=

F

Z

up

a

.

Obvious

295

.

Cor

0

is dual of Cor.

Theorem

296

.

Cor

a

v

a

whenever Cor

a

exists for any element

a

of a filtered

filtrator.

Proof.

Cor

a

=

d

Z

up

a

v

d

A

up

a

=

a

.

Corollary

297

.

Cor

a

down

a

whenever Cor

a

exists for any element

a

of a

filtered filtrator.

Theorem

298

.

Cor

0

a

v

a

whenever Cor

0

a

exists for any element

a

of a filtrator

with join-closed core.

Proof.

Cor

0

a

=

F

Z

down

a

=

F

A

down

a

v

a

.

Corollary

299

.

Cor

0

a

down

a

whenever Cor

0

a

exists for any element

a

of

a filtrator with join-closed core.

Proposition

300

.

Cor

0

a

v

Cor

a

whenever both Cor

a

and Cor

0

a

exist for

any element

a

of a filtrator with join-closed core.