 Thus we have proved

f

g

t

f

h

w

f

(

g

t

h

)

. But obviously

f

(

g

t

h

)

w

f

g

and

f

(

g

t

h

)

w

f

h

and so

f

(

g

t

h

)

w

f

g

t

f

h

. Thus

f

(

g

t

h

) =

f

g

t

f

h

.

Theorem 7.16.

Let

A

,

B

,

C

be sets,

f

2

RLD

(

A

;

B

)

,

g

2

RLD

(

B

;

C

)

,

h

2

RLD

(

A

;

C

)

. Then

g

f

/

h

,

g

/

h

f

¡

1

:

Proof.

g

f

/

h

,

d

f"

RLD

(

G

F

)

j

F

2

xyGR

f ; G

2

xyGR

g

g u

d

h"

RLD

i

xyGR

h

=

/ 0

RLD

(

A

;

C

)

,

d

f"

RLD

((

G

F

)

u

H

)

j

F

2

xyGR

f ; G

2

xyGR

g; H

2

xyGR

h

g

=

/ 0

RLD

(

A

;

C

)

, 8

F

2

xyGR

f ;

G

2

xyGR

g; H

2

xyGR

h

:

"

RLD

((

G

F

)

u

H

) =

/ 0

RLD

(

A

;

C

)

, 8

F

2

xyGR

f ; G

2

xyGR

g; H

2

xyGR

h

:

G

F

/

H

(used properties of generalized lter bases).

Similarly

g

/

h

f

¡

1

, 8

F

2

xyGR

f ; G

2

xyGR

g; H

2

xyGR

h

:

G

/

H

F

¡

1

.

Thus

g

f

/

h

,

g

/

h

f

¡

1

because

G

F

/

H

,

G

/

H

F

¡

1

by proposition

3.70

.

Theorem 7.17.

For every composable reloids

f

and

g

1.

g

f

=

F

f

g

F

j

F

2

atoms

f

g

.

2.

g

f

=

F

f

G

f

j

G

2

atoms

g

g

.

Proof.

We will prove only the rst as the second is dual.

F

f

g

F

j

F

2

atoms

f

g

=

g

f

, 8

x

2

RLD

(

Src

f

;

Dst

g

): (

x

/

g

f

,

x

/

F

f

g

F

j

F

2

atoms

f

g

)

( 8

x

2

RLD

(

Src

f

;

Dst

g

): (

x

/

g

f

, 9

F

2

atoms

f

:

x

/

g

F

)

, 8

x

2

RLD

(

Src

f

;

Dst

g

):

(

g

¡

1

x

/

f

, 9

F

2

atoms

f

:

g

¡

1

x

/

F

)

what is obviously true.

Corollary 7.18.

If

f

and

g

are composable reloids, then

g

f

=

G

f

G

F

j

F

2

atoms

f ; G

2

atoms

g

g

:

Proof.

g

f

=

F

f

g

F

j

F

2

atoms

f

g

=

F

f

F

f

G

F

j

G

2

atoms

g

g j

F

2

atoms

f

g

=

F

f

G

F

j

F

2

atoms

f ; G

2

atoms

g

g

.

7.3 Direct product of lters

Denition 7.19.

Reloidal product

of lters

A

and

B

is dened by the formula

RLD

B

=

def

l

"

RLD

(

Base

(

A

);

Base

(

B

))

(

A

B

)

j

A

2 A

; B

2 B

:

Obvious 7.20.

"

U

A

RLD

"

V

B

=

"

RLD

(

U

;

V

)

(

A

B

)

for every sets

A

U

,

B

V

.

Theorem 7.21.

RLD

B

=

F

f

a

RLD

b

j

a

2

atoms

A

; b

2

atoms

Bg

for every lters

A

,

B

.

Proof.

Obviously

RLD

B w

G

f

a

RLD

b

j

a

2

atoms

A

; b

2

atoms

Bg

:

Reversely, let

K

2

GR

G

f

a

RLD

b

j

a

2

atoms

A

; b

2

atoms

Bg

:

Then

K

2

GR

(

a

RLD

b

)

for every

a

2

atoms

A

,

b

2

atoms

B

;

K

X

a

Y

b

for some

X

a

2

a

,

Y

b

2

b

;

K

S

f

X

a

Y

b

j

a

2

atoms

A

; b

2

atoms

Bg

=

S

f

X

a

j

a

2

atoms

Ag

S

f

Y

b

j

b

2

atoms

Bg

A

B

where

A

2 A

,

B

2 B

;

K

2

GR

(

RLD

B

)

.

Theorem 7.22.

If

A

0

;

A

1

2

F

(

A

)

,

B

0

;

B

1

2

F

(

B

)

for some sets

A

,

B

then

(

A

0

RLD

B

0

)

u

(

A

1

RLD

B

1

) = (

A

0

u A

1

)

RLD

(

B

0

u B

1

)

:

7.3 Direct product of filters

123