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Theorem 8.3.

(

FCD

)

f

=

d

h"

FCD

i

xyGR

f

for every reloid

f

.

Proof.

Let

a

be an ultralter on Src

f

.

h

(

FCD

)

f

i

a

=

d

fh"

FCD

F

i

a

j

F

2

xyGR

f

g

by the denition of

(

FCD

)

.

h

d

h"

FCD

i

xyGR

f

i

a

=

d

fh"

FCD

F

i

a

j

F

2

xyGR

f

g

by theorem

6.68

.

So

h

(

FCD

)

f

i

a

=

h

d

h"

FCD

i

xyGR

f

i

a

for every ultralter

a

.

Lemma 8.4.

For every two lter bases

S

and

T

of morphisms Rel

(

U

;

V

)

and every set

A

U

l

h"

RLD

i

S

=

l

h"

RLD

i

T

)

l

f"

V

h

F

i

A

j

F

2

S

g

=

l

f"

V

h

G

i

A

j

G

2

T

g

:

Proof.

Let

d

h"

RLD

i

S

=

d

h"

RLD

i

T

.

First let prove that

fh

F

i

A

j

F

2

S

g

is a lter base. Let

X ; Y

2fh

F

i

A

j

F

2

S

g

. Then

X

=

h

F

X

i

A

and

Y

=

h

F

Y

i

A

for some

F

X

; F

Y

2

S

. Because

S

is a lter base, we have

S

3

F

Z

v

F

X

u

F

Y

. So

h

F

Z

i

A

v

X

u

Y

and

h

F

Z

i

A

2 fh

F

i

A

j

F

2

S

g

. So

fh

F

i

A

j

F

2

S

g

is a lter base.

Suppose

X

2

d

f"

V

h

F

i

A

j

F

2

S

g

. Then there exists

X

0

2 fh

F

i

A

j

F

2

S

g

where

X

w

X

0

because

fh

F

i

A

j

F

2

S

g

is a lter base. That is

X

0

=

h

F

i

A

for some

F

2

S

. There exists

G

2

T

such

that

G

v

F

because

T

is a lter base. Let

Y

0

=

h

G

i

A

. We have

Y

0

v

X

0

v

X

;

Y

0

2 fh

G

i

A

j

G

2

T

g

;

Y

0

2

d

f"

V

h

G

i

A

j

G

2

T

g

;

X

2

d

f"

V

h

G

i

A

j

G

2

T

g

. The reverse is symmetric.

Lemma 8.5.

f

G

F

j

F

2

GR

f ; G

2

GR

g

g

is a lter base for every reloids

f

and

g

.

Proof.

Let denote

D

=

f

G

F

j

F

2

GR

f ; G

2

GR

g

g

. Let

A

2

D

^

B

2

D

. Then

A

=

G

A

F

A

^

B

=

G

B

F

B

for some

F

A

; F

B

2

GR

f

,

G

A

; G

B

2

GR

g

. So

A

\

B

(

G

A

\

G

B

)

(

F

A

\

F

B

)

2

D

because

F

A

\

F

B

2

GR

f

and

G

A

\

G

B

2

GR

g

.

Theorem 8.6.

(

FCD

)(

g

f

) = ((

FCD

)

g

)

((

FCD

)

f

)

for every composable reloids

f

and

g

.

Proof.

h

(

FCD

)(

g

f

)

i

X

=

l

f"

Dst

g

h

H

i

X

j

H

2

GR

(

g

f

)

g

=

l

"

Dst

g

h

H

i

X

j

H

2

GR

l

f"

RLD

(

G

F

)

j

F

2

xyGR

f ; G

2

xyGR

g

g

 

:

Obviously

l

f"

RLD

(

G

F

)

j

F

2

xyGR

f ; G

2

xyGR

g

g

=

l

h"

RLD

i

xyGR

l

f"

RLD

(

G

F

)

j

F

2

xyGR

f ; G

2

xyGR

g

g

;

from this by lemma

8.4

(taking into account that

f

G

F

j

F

2

GR

f ; G

2

GR

g

g

and

GR

l

f"

RLD

(

G

F

)

j

F

2

xyGR

f ; G

2

xyGR

g

g

are lter bases)

l

"

Dst

g

h

H

i

X

j

H

2

GR

l

f"

RLD

(

G

F

)

j

F

2

xyGR

f ; G

2

xyGR

g

g

 

=

l

f"

Dst

g

h

G

F

i

X

j

F

2

GR

f ; G

2

GR

g

g

:

On the other side

h

((

FCD

)

g

)

((

FCD

)

f

)

i

X

=

h

(

FCD

)

g

ih

(

FCD

)

f

i

X

=

h

(

FCD

)

g

i

l

f"

Dst

g

h

F

i

X

j

F

2

xyGR

f

g

=

l

h"

FCD

G

i

l

f"

Dst

g

h

F

i

X

j

F

2

xyGR

f

g j

G

2

xyGR

g

 

:

Let's prove that

fh

F

i

X

j

F

2

xyGR

f

g

is a lter base. If

A; B

2 fh

F

i

X

j

F

2

xyGR

f

g

then

A

=

h

F

1

i

X

,

B

=

h

F

2

i

X

where

F

1

; F

2

2

xyGR

f

.

A

\

B

 h

F

1

u

F

2

i

X

2 fh

F

i

X

j

F

2

xyGR

f

g

. So

fh

F

i

X

j

F

2

xyGR

f

g

is really a lter base.

132

Relationships between funcoids and reloids