If

X

2 ff

0

gg

,

Y

2 ff

1

gg

then

X

t

Y

2

/

ff

0

g

;

f

0

;

1

gg

??
That

X

i

=

; )

X

2

/ (

val

f

)

k

L

is obvious. So

f

is a pre-multifuncoid.

??

Conjecture 29.

If

a

is a completary multifuncoid and

Dst

f

i

is a starrish poset for every

i

2

n

then

StarComp

(

a

;

f

)

is a completary multifuncoid.

Proof.

Let

8

K

2

Q

form

f

: (

K

w

L

0

^

K

w

L

1

)

K

2

StarComp

(

a

;

f

))

that is

8

K

2

Q

form

f

:

¡

K

w

L

0

^

K

w

L

1

) 9

y

2

Q

i

2

n

atoms

A

i

: (

8

i

2

n

:

y

i

[

f

i

]

K

i

^

y

2

a

)

that is

8

K

2

Q

form

f

9

y

2

Q

i

2

n

atoms

A

i

: (

K

w

L

0

^

K

w

L

1

)

(

8

i

2

n

:

y

i

[

f

i

]

K

i

^

y

2

a

))

that is

8

K

2

Q

form

f

9

y

2

Q

i

2

n

atoms

A

i

: ((

K

w

L

0

^

K

w

L

1

) 8

i

2

n

:

y

i

[

f

i

]

K

i

)

^

y

2

a

)

that is??

8

K

2

Q

form

f

9

y

2

Q

i

2

n

atoms

A

i

: (

9

c

2 f

0

;

1

g

n

8

i

2

n

:

y

i

[

f

i

]

L

c

(

i

)

i

^

y

2

a

)

that is ??

Conjecture 30.

Q

(

D

)

F

is a pre-multifuncoid if every

F

i

is a pre-multifuncoid.

Proof.

Let

X ; Y

2

form

Q

(

D

)

F

(

i

;

j

)

.

8

Z

2

form

Q

(

D

)

F

(

i

;

j

)

:

Z

X

^

Z

Y

)

Z

2

val

Q

(

D

)

F

(

i

;

j

)

L

, 8

Z

2

form

Q

(

D

)

F

(

i

;

j

)

: (

Z

X

^

Z

Y

) 9

K

2

(

form

F

i

)

j

(

arity

F

i

)

nf

j

g

:

Z

2

(

val

F

j

)

K

)

, 8

Z

2

form

Q

(

D

)

F

(

i

;

j

)

: (

9

K

2

(

form

F

i

)

j

(

arity

F

i

)

nf

j

g

: (

Z

X

^

Z

Y

)

Z

2

(

val

F

j

)

K

))

,

??

, 8

Z

2

form

Q

(

D

)

F

(

i

;

j

)

: (

9

K

2

(

form

F

i

)

j

(

arity

F

i

)

nf

j

g

: (

X

2

(

val

F

j

)

K

_

Y

2

(

val

F

j

)

K

))

,

8

Z

2

form

Q

(

D

)

F

(

i

;

j

)

: (

9

K

2

(

form

F

i

)

j

(

arity

F

i

)

nf

j

g

:

X

2

(

val

F

j

)

K

_ 9

K

2

(

form

F

i

)

j

(

arity

F

i

)

nf

j

g

:

Y

2

(

val

F

j

)

K

)

,

Let

f

is a funcoid.

Then there exists a reloid

g

such that ??

=============

(

RLD

)

in

f

=

a

RLD

b

j

a

2

atoms

1

F

(

Src

f

)

; b

2

atoms

1

F

(

Dst

f

)

; a

FCD

b

f

(

RLD

)

in

(

g

f

) =

a

RLD

b

j

a

2

atoms

1

F

(

Src

f

)

; b

2

atoms

1

F

(

Dst

f

)

; a

FCD

b

g

f

=

a

RLD

b

j

a

2

atoms

1

F

(

Src

f

)

; b

2

atoms

1

F

(

Dst

f

)

; a

RLD

b

(

RLD

)

in

(

g

f

)

(

RLD

)

in

(

FCD

)((

RLD

)

in

g

(

RLD

)

in

f

) = (

RLD

)

in

((

FCD

)(

RLD

)

in

g

(

FCD

)(

RLD

)

in

f

) = (

RLD

)

in

(

g

f

)

Lemma 31.

8

Y

2

up

h

f

i

X

9

F

2

up

f

:

h

F

i

X

Y

for every funcoid

f

.

Proof.

??

(

RLD

)

in

(

g

f

) =

Theorem 32.

g

f

=

"

FCD

(

Src

f

;

Dst

g

)

(

G

F

)

j

F

2

up

f ; G

2

up

g

Proof.

It's enough?? to prove that

8

H

2

up

(

g

f

)

9

F

2

up

f ; G

2

up

g

:

H

G

F

.

X

[

g

f

]

Y

, h

f

i

X

/

h

g

¡

1

i

Y

, 8

X

0

2

up

h

f

i

X ; Y

0

2

up

h

g

¡

1

i

Y

:

X

0

/

Y

0

, 9

F

2

up

f ;

G

2

up

g

:

h

F

i

X

/

h

G

i

Y

, 9

F

2

up

f ; G

2

up

g

:

X

[

G

F

]

Y

(used the lemma).

Let

H

2

up

(

g

f

)

. Then

X

[

H

]

Y

)

X

[

g

f

]

Y

) 9

F

2

up

f ; G

2

up

g

:

X

[

G

F

]

Y

for every

X

,

Y

. Thus ??(it does not work because

F

amd

G

depend on

X

and

Y

).

Lemma 33.

f

r

=

T

f

f

"

FCD

R

j

R

2

up

r

g

Proof.

Obviously

f

r

T

f

f

"

FCD

R

j

R

2

up

r

g

.

h

f

r

i

X

??

T

fh

f

ih

R

i

X

j

R

2

up

r

g

=

T

fh

f

ih"

FCD

R

i

X

j

R

2

up

r

g

=

T

fh

f

"

FCD

R

i

X

j

R

2

up

r

g  h

T

f

f

"

FCD

R

j

R

2

up

r

gi

X

10

Section 6