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Definition 39

I call a filtrator with

meet-closed core

such filtrator

(

A

;

Z

)

that

T

Z

S

=

T

A

S

whenever

T

Z

S

exists for

S

∈ P

Z

.

Definition 40

I call a filtrator with

finitely join-closed core

such filtrator

(

A

;

Z

)

that

a

Z

b

=

a

A

b

whenever

a

Z

b

exists for

a, b

Z

.

Definition 41

I call a filtrator with

finitely meet-closed core

such filtrator

(

A

;

Z

)

that

a

Z

b

=

a

A

b

whenever

a

Z

b

exists for

a, b

Z

.

Definition 42

Filtered filtrator

is a filtrator

(

A

;

Z

)

such that

a

A

:

a

=

T

A

up

a

.

Definition 43

Prefiltered filtrator

is a filtrator

(

A

;

Z

)

such that “

up

” is in-

jective.

Definition 44

Semifiltered filtrator

is a filtrator

(

A

;

Z

)

such that

a, b

A

: (up

a

up

b

a

b

)

.

Obvious 9

Every filtered filtrator is semifiltered.

Every semifiltered filtrator is prefiltered.

Obvious 10

up

” is a straight map from

A

to the dual of the poset

P

Z

if

(

A

;

Z

)

is a semifiltered filtrator.

Theorem 23

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

Proof

Let (

A

;

Z

) be a semifiltered filtrator. Let

S

∈ P

Z

and

S

Z

S

is defined.

We need to prove

S

A

S

=

S

Z

S

. That

S

Z

S

is an upper bound for

S

is obvious.

Let

a

A

be an upper bound for

S

. Enough to prove that

S

Z

S

a

. Really,

c

up

a

c

a

⇒ ∀

x

S

:

c

x

c

[

Z

S

c

up

[

Z

S

;

so up

a

up

S

Z

S

and thus

a

S

Z

S

because it is semifiltered.

4.1. Core part

Definition 45

The

core part

of an element

a

A

is

Cor

a

=

T

Z

up

a

.

Definition 46

The

dual core part

of an element

a

A

is

Cor

a

=

S

Z

down

a

.

Obvious 11

Cor

is dual of

Cor

.

Theorem 24

Cor

a

a

whenever

Cor

a

exists for any element

a

of a filtered

filtrator.

20