Proof.

(

A

0

RLD

B

0

)

u

(

A

1

RLD

B

1

)

=

f"

RLD

(

P

u

Q

)

j

P

2

xyGR

(

A

0

RLD

B

0

)

; Q

2

xyGR

(

A

1

RLD

B

1

)

g

=

"

RLD

(

A

;

B

)

((

A

0

B

0

)

\

(

A

1

B

1

))

j

A

0

2 A

0

; B

0

2 B

0

; A

1

2 A

1

; B

1

2 B

1

=

"

RLD

(

A

;

B

)

((

A

0

\

A

1

)

(

B

0

\

B

1

))

j

A

0

2 A

0

; B

0

2 B

0

; A

1

2 A

1

; B

1

2 B

1

=

"

RLD

(

A

;

B

)

(

K

L

)

j

K

2 A

0

u A

1

; L

2 B

0

u B

1

= (

A

0

u A

1

)

RLD

(

B

0

u B

1

)

:

Theorem 7.23.

If

S

2

P

(

F

(

A

)

F

(

B

))

for some sets

A

,

B

then

l

fA

RLD

B j

(

A

;

B

)

2

S

g

=

l

dom

S

RLD

l

im

S:

Proof.

Let

P

=

d

dom

S

,

Q

=

d

im

S

;

l

=

d

fA

RLD

B j

(

A

;

B

)

2

S

g

.

RLD

Q v

l

is obvious.

Let

F

2

GR

(

RLD

Q

)

. Then there exist

P

2 P

and

Q

2 Q

such that

F

P

Q

.

P

=

P

1

\

:::

\

P

n

where

P

i

2

dom

S

and

Q

=

Q

1

\

:::

\

Q

m

where

Q

j

2

im

S

.

P

Q

=

T

i; j

(

P

i

Q

j

)

.

P

i

Q

j

2

GR

(

RLD

B

)

for some

(

A

;

B

)

2

S

.

P

Q

=

T

i; j

(

P

i

Q

j

)

2

GR

l

. So

F

2

GR

l

.

Corollary 7.24.

d

hA

RLD

i

T

=

RLD

d

T

if

A

is a lter and

T

is a set of lters with common

base.

Proof.

Take

S

=

fAg

T

where

T

is a set of lters.

Then

d

fA

RLD

B j B 2

T

g

=

RLD

d

T

that is

d

hA

RLD

i

T

=

RLD

d

T

.

Denition 7.25.

I will call a reloid

convex

i it is a join of direct products.

7.4 Restricting reloid to a lter. Domain and image

Denition 7.26.

Identity reloid

for a set

A

is dened by the formula id

RLD

(

A

)

=

"

RLD

(

A

;

A

)

id

A

.

Obvious 7.27.

¡

id

RLD

(

A

)

¡

1

=

id

RLD

(

A

)

.

Denition 7.28.

I dene

restricting

a reloid

f

to a lter

A

as

f

j

A

=

f

u

¡

RLD

1

F

(

Dst

f

)

.

Denition 7.29.

Domain

and

image

of a reloid

f

are dened as follows:

dom

f

=

l

h"

Src

f

ih

dom

i

GR

f

;

im

f

=

l

h"

Dst

f

ih

im

i

GR

f :

Proposition 7.30.

f

v A

RLD

B,

dom

f

v A^

im

f

v B

for every reloid

f

and lters

A 2

F

(

Src

f

)

,

B 2

F

(

Dst

f

)

.

Proof.

)

.

It follows from dom

(

RLD

B

)

v A ^

im

(

RLD

B

)

v B

.

(

.

dom

f

v A , 8

A

2 A9

F

2

GR

f

:

dom

F

A

. Analogously

im

f

v B , 8

B

2 B9

G

2

GR

f

:

im

G

B

.

Let dom

f

v A ^

im

f

v B

,

A

2 A

,

B

2 B

. Then there exist

F ; G

2

GR

f

such that

dom

F

A

^

im

G

B

. Consequently

F

\

G

2

GR

f

, dom

(

F

\

G

)

A

, im

(

F

\

G

)

B

that

is

F

\

G

A

B

. So there exists

H

2

GR

f

such that

H

A

B

for every

A

2 A

,

B

2 B

.

So

f

v A

RLD

B

.

Denition 7.31.

I call

restricted identity reloid

for a lter

A

the reloid

id

A

RLD

=

def

¡

id

RLD

(

Base

(

A

))

j

A

:

124

Reloids