7.2. COMPOSITION OF RELOIDS

134

Theorem

743

.

(

h

g

)

f

=

h

(

g

f

) for every composable reloids

f

,

g

,

h

.

Proof.

For two nonempty collections A and B of sets I will denote

A

B

⇔ ∀

K

A

L

B

:

L

K

∧ ∀

K

B

L

A

:

L

K.

It is easy to see that

is a transitive relation.

I will denote

B

A

=

n

L

K

K

A,L

B

o

.

Let first prove that for every nonempty collections of relations

A

,

B

,

C

A

B

A

C

B

C.

Suppose

A

B

and

P

A

C

that is

K

A

and

M

C

such that

P

=

K

M

.

K

0

B

:

K

0

K

because

A

B

. We have

P

0

=

K

0

M

B

C

. Obviously

P

0

P

. So for every

P

A

C

there exists

P

0

B

C

such that

P

0

P

; the vice

versa is analogous. So

A

C

B

C

.

GR((

h

g

)

f

)

GR(

h

g

)

GR

f

, GR(

h

g

)

(GR

h

)

(GR

g

). By proven

above GR((

h

g

)

f

)

(GR

h

)

(GR

g

)

(GR

f

).

Analogously GR(

h

(

g

f

))

(GR

h

)

(GR

g

)

(GR

f

).

So GR(

h

(

g

f

))

GR((

h

g

)

f

) what is possible only if GR(

h

(

g

f

)) =

GR((

h

g

)

f

). Thus (

h

g

)

f

=

h

(

g

f

).

Theorem

744

.

For every reloid

f

:

1

.

f

f

=

d

n

RLD

(

F

F

)

F

xyGR

f

o

if Src

f

= Dst

f

;

2

.

f

1

f

=

d

n

RLD

(

F

1

F

)

F

xyGR

f

o

;

3

.

f

f

1

=

d

n

RLD

(

F

F

1

)

F

xyGR

f

o

.

Proof.

I will prove only

1

and

2

because

3

is analogous to

2

.

1

It’s enough to show that

F, G

xyGR

f

H

xyGR

f

:

H

H

v

G

F

.

To prove it take

H

=

F

u

G

.

2

It’s enough to show that

F, G

xyGR

f

H

xyGR

f

:

H

1

H

v

G

1

F

. To prove it take

H

=

F

u

G

. Then

H

1

H

= (

F

u

G

)

1

(

F

u

G

)

v

G

1

F

.

Theorem

745

.

For every sets

A

,

B

,

C

if

g, h

RLD

(

A

;

B

) then

1

.

f

(

g

t

h

) =

f

g

t

f

h

for every

f

RLD

(

B

;

C

);

2

. (

g

t

h

)

f

=

g

f

t

h

f

for every

f

RLD

(

C

;

A

).

Proof.

We’ll prove only the first as the second is dual.

By the infinite distributivity law for filters we have

f

g

t

f

h

=

l

RLD

(

F

G

)

F

xyGR

f, G

xyGR

g

t

l

RLD

(

F

H

)

F

xyGR

f, H

xyGR

h

=

l

RLD

(

F

1

G

)

t ↑

RLD

(

F

2

H

)

F

1

, F

2

xyGR

f, G

xyGR

g, H

xyGR

h

=

l

RLD

((

F

1

G

)

t

(

F

2

H

))

F

1

, F

2

xyGR

f, G

xyGR

g, H

xyGR

h

.