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8.4. FUNCOIDAL RELOIDS

155

Proof.

Cor(

FCD

)

g

=

G

FCD

{

(

x

;

y

)

}

FCD

{

(

x

;

y

)

} v

(

FCD

)

g

=

G

FCD

{

(

x

;

y

)

}

RLD

{

(

x

;

y

)

} v

g

=

G

(

FCD

)

RLD

{

(

x

;

y

)

}

RLD

{

(

x

;

y

)

} v

g

=

(

FCD

)

G

RLD

{

(

x

;

y

)

}

RLD

{

(

x

;

y

)

} v

g

=

(

FCD

) Cor

g.

Conjecture

841

.

1

. Cor(

RLD

)

in

g

= (

RLD

)

in

Cor

g

;

2

. Cor(

RLD

)

out

g

= (

RLD

)

out

Cor

g

.

8.4. Funcoidal reloids

Definition

842

.

I call

funcoidal

such a reloid

ν

that

X ×

RLD

Y ⇒

∃X

0

F

(Base(

X

))

\{⊥}

,

Y

0

F

(Base(

Y

))

\{⊥}

: (

X

0

v X ∧Y

0

v Y∧X

0

×

RLD

Y

0

v

ν

)

for every

X ∈

F

(Src

ν

),

Y ∈

F

(Dst

ν

).

Proposition

843

.

A reloid

ν

is funcoidal iff

x

×

RLD

y

6

ν

x

×

RLD

y

v

ν

for

every ultrafilters

x

and

y

on respective sets.

Proof.

.

x

×

RLD

y

6

ν

⇒ ∃X

0

atoms

x,

Y

0

atoms

y

:

X

0

×

RLD

Y

0

v

ν

x

×

RLD

y

v

ν

.

.

X ×

RLD

Y ⇒

x

atoms

X

, y

atoms

Y

:

x

×

RLD

y

6

ν

x

atoms

X

, y

atoms

Y

:

x

×

RLD

y

v

ν

∃X

0

F

(Base(

X

))

\{⊥}

,

Y

0

F

(Base(

Y

))

\{⊥}

: (

X

0

v X ∧Y

0

v Y∧X

0

×

RLD

Y

0

v

ν

)

.

Proposition

844

.

(

RLD

)

in

(

FCD

)

f

=

F

n

a

×

RLD

b

a

atoms

F

(Src

ν

)

,b

atoms

F

(Dst

ν

)

,a

×

RLD

b

6

f

o

.

Proof.

(

RLD

)

in

(

FCD

)

f

=

G

a

×

RLD

b

a

atoms

F

(Src

ν

)

, b

atoms

F

(Dst

ν

)

, a

×

FCD

b

v

(

FCD

)

f

=

G

a

×

RLD

b

a

atoms

F

(Src

ν

)

, b

atoms

F

(Dst

ν

)

, a

[(

FCD

)

f

]

b

=

G

a

×

RLD

b

a

atoms

F

(Src

ν

)

, b

atoms

F

(Dst

ν

)

, a

×

RLD

b

6

f

.