0

RLD

(

A

;

C

)

⇔ ∀

F

up

f, G

up

g, H

up

h

:

G

F

6≍

H

(used properties

of generalized filter bases).

Similarly

g

6≍

h

f

1

⇔ ∀

F

up

f, G

up

g, H

up

h

:

G

6≍

H

F

1

.

Thus

g

f

6≍

h

g

6≍

h

f

1

because

G

F

6≍

H

G

6≍

H

F

1

by the

proposition 1.

4.2

Direct product of filter objects

Definition 46

Reloidal product

of filter objects

A

and

B

is defined by the

formula

A ×

RLD

B

def

=

\ n

RLD

(Base(

A

);Base(

B

))

(

A

×

B

)

|

A

up

A

, B

up

B

o

.

Obvious 20.

U

A

×

RLD

V

B

=

RLD

(

U

;

V

)

(

A

×

B

) for every small sets

A

U

and

B

V

.

Theorem 46

A ×

RLD

B

=

a

×

RLD

b

|

a

atoms

A

, b

atoms

B

for ev-

ery filter objects

A

,

B

.

Proof

Obviously

A ×

RLD

B ⊇

a

×

RLD

b

|

a

atoms

A

, b

atoms

B

.

Reversely, let

K

up

a

×

RLD

b

|

a

atoms

A

, b

atoms

B

.

Then

K

up(

a

×

RLD

b

) for every

a

atoms

A

, b

atoms

B

;

K

X

a

×

Y

b

for

some

X

a

up

a

,

Y

b

up

b

;

K

S

{

X

a

×

Y

b

|

a

atoms

A

, b

atoms

B}

=

S

{

X

a

|

a

atoms

A} ×

S

{

Y

b

|

b

atoms

B} ⊇

A

×

B

where

A

up

A

,

B

up

B

;

K

up(

A ×

RLD

B

).

Theorem 47

If

A

0

,

A

1

F

(

A

)

,

B

0

,

B

1

F

(

B

)

for some small sets

A

,

B

then

(

A

0

×

RLD

B

0

)

(

A

1

×

RLD

B

1

) = (

A

0

∩ A

1

)

×

RLD

(

B

0

∩ B

1

)

.

Proof

(

A

0

×

RLD

B

0

)

(

A

1

×

RLD

B

1

)

=

RLD

(

A

;

B

)

(

P

Q

)

|

P

up(

A

0

×

RLD

B

0

)

, Q

up(

A

1

×

RLD

B

1

)

=

RLD

(

A

;

B

)

((

A

0

×

B

0

)

(

A

1

×

B

1

))

|

A

0

up

A

0

, B

0

up

B

0

, A

1

up

A

1

, B

1

up

B

1

=

RLD

(

A

;

B

)

((

A

0

A

1

)

×

(

B

0

B

1

))

|

A

0

up

A

0

, B

0

up

B

0

, A

1

up

A

1

, B

1

up

B

1

=

RLD

(

A

;

B

)

(

K

×

L

)

|

K

up(

A

0

∩ A

1

)

, L

up(

B

0

∩ B

1

)

=

(

A

0

∩ A

1

)

×

RLD

(

B

0

∩ B

1

)

.

42