background image

Let’s denote

W

=

Y ∩ h

f

i

X

|

X

up

X

 

. We will prove that

W

is a generalized filter base. To prove this it is enough to show that

V

=

h

f

i

X

|

X

up

X

 

is a generalized filter base.

Let

P

,

Q ∈

V

. Then

P

=

h

f

i

A

,

Q

=

h

f

i

B

where

A, B

up

X

;

A

B

up

X

and

R ⊆ P ∩ Q

for

R

=

h

f

i

(

A

B

)

V

. So

V

is a generalized filter

base and thus

W

is a generalized filter base.

0

F

(Dst

f

)

/

W

T

W

6

= 0

F

(Dst

f

)

by the corollary of the theorem 1. That

is

X

up

X

:

Y ∩ h

f

i

X

6

= 0

F

(Dst

f

)

⇔ Y ∩

h

f

i

up

X 6

= 0

F

(Dst

f

)

.

Comparing with the above,

Y ∩ h

f

i X 6

= 0

F

(Dst

f

)

⇔ Y ∩

h

f

i

up

X 6

=

0

F

(Dst

f

)

. So

h

f

i X

=

h

f

i

up

X

because the lattice of filter objects is sepa-

rable.

Proposition 10

For every

f

FCD

(

A

;

B

)

we have (for every

I, J

P

A

)

h

f

i

= 0

F

(

B

)

,

h

f

i

(

I

J

) =

h

f

i

I

∪ h

f

i

J

and

¬

(

I

[

f

]

)

, I

J

[

f

]

K

I

[

f

]

K

J

[

f

]

K

(for every

I, J

P

A

,

K

P

B

)

,

¬

(

[

f

]

I

)

,

K

[

f

]

I

J

K

[

f

]

I

K

[

f

]

J

(for every

I, J

P

B

,

K

P

A

).

Proof

h

f

i

=

h

f

i ↑

A

=

h

f

i

0

F

(

A

)

= 0

F

(

B

)

;

h

f

i

(

I

J

) =

h

f

i ↑

A

(

I

J

) =

h

f

i ↑

A

I

∪ ↑

A

J

=

h

f

i ↑

A

I

∪ h

f

i ↑

A

J

=

h

f

i

I

∪ h

f

i

J

.

I

[

f

]

∅ ⇔

0

F

(

B

)

6≍ h

f

i ↑

A

I

0;

I

J

[

f

]

K

⇔↑

A

(

I

J

) [

f

]

B

K

⇔↑

B

K

6≍ h

f

i

(

I

J

)

⇔↑

B

K

6≍ h

f

i

I

∪ h

f

i

J

⇔↑

B

K

6≍ h

f

i

I

∨ ↑

B

K

6≍ h

f

i

J

I

[

f

]

K

J

[

f

]

K

.

The rest follows from symmetry.

Theorem 7

Fix small sets

A

and

B

. Let

L

F

=

λf

FCD

(

A

;

B

) :

h

f

i

and

L

R

=

λf

FCD

(

A

;

B

) :[

f

]

.

1.

L

F

is a bijection from the set

FCD

(

A

;

B

)

to the set of functions

α

F

(

B

)

P

A

that obey the conditions (for every

I, J

P

A

)

α

= 0

F

(

B

)

,

α

(

I

J

) =

αI

αJ.

(1)

For such

α

it holds (for every

X ∈

F

(

A

)

)

L

1

F

α

X

=

\

h

α

i

up

X

.

(2)

2.

L

R

is a bijection from the set

FCD

(

A

;

B

)

to the set of binary relations

δ

P

(

P

A

×

P

B

)

that obey the conditions

¬

(

I δ

)

, I

J δ K

I δ K

J δ K

(for every

I, J

P

A

,

K

P

B

)

,

¬

(

δ I

)

, K δ I

J

K δ I

K δ J

(for every

I, J

P

B

,

K

P

A

).

(3)

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