7.2. COMPOSITION OF RELOIDS

135

Obviously

l

RLD

((

F

1

G

)

t

(

F

2

H

))

F

1

, F

2

xyGR

f, G

xyGR

g, H

xyGR

h

w

l

RLD

(((

F

1

u

F

2

)

G

)

t

((

F

1

u

F

2

)

H

))

F

1

, F

2

xyGR

f, G

xyGR

g, H

xyGR

h

=

l

RLD

((

F

G

)

t

(

F

H

))

F

xyGR

f, G

xyGR

g, H

xyGR

h

=

l

RLD

(

F

(

G

t

H

))

F

xyGR

f, G

xyGR

g, H

xyGR

h

.

Because

G

xyGR

g

H

xyGR

h

G

t

H

xyGR(

g

t

h

) we have

l

RLD

(

F

(

G

t

H

))

F

xyGR

f, G

xyGR

g, H

xyGR

h

w

l

RLD

(

F

K

)

F

xyGR

f, K

xyGR(

g

t

h

)

=

f

(

g

t

h

)

.

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

746

.

Let

A

,

B

,

C

be sets,

f

RLD

(

A

;

B

),

g

RLD

(

B

;

C

),

h

RLD

(

A

;

C

). Then

g

f

6

h

g

6

h

f

1

.

Proof.

g

f

6

h

l

RLD

(

G

F

)

F

xyGR

f, G

xyGR

g

u

l

RLD

xyGR

h

6

=

RLD

(

A

;

C

)

l

RLD

((

G

F

)

u

H

)

F

xyGR

f, G

xyGR

g, H

xyGR

h

6

=

RLD

(

A

;

C

)

F

xyGR

f, G

xyGR

g, H

xyGR

h

:

RLD

((

G

F

)

u

H

)

6

=

RLD

(

A

;

C

)

F

xyGR

f, G

xyGR

g, H

xyGR

h

:

G

F

6

H

(used properties of generalized filter bases).

Similarly

g

6

h

f

1

⇔ ∀

F

xyGR

f, G

xyGR

g, H

xyGR

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

proposition

229

.

Theorem

747

.

For every composable reloids

f

and

g

1

.

g

f

=

F

n

g

F

F

atoms

f

o

.

2

.

g

f

=

F

n

G

f

G

atoms

g

o

.

Proof.

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