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9.5. EMBEDDING RELOIDS INTO FUNCOIDS

160

Proof.

h

ρ

(

g

f

)

i

x

=

g

f

x

=

h

ρg

ih

ρf

i

x

= (

h

ρg

i ◦ h

ρf

i

)

x

. Thus

h

ρ

(

g

f

)

i

=

h

ρg

i ◦ h

ρf

i

=

h

(

ρg

)

(

ρf

)

i

and so

ρ

(

g

f

) = (

ρg

)

(

ρf

).

Theorem

862

.

ρ

F

F

=

F

h

ρ

i

F

for a set

F

of reloids.

Proof.

It’s enough to prove

h

ρ

F

F

i

X

=

F

h

ρ

i

F

X

for a set

X

.

Really,

D

ρ

G

F

E

X

=

D

ρ

G

F

E

X

=

G

F

◦ ↑

X

=

G

f

◦ ↑

X

f

F

=

G

h

ρf

i ↑

X

f

F

=

G

ρf

f

F

X

=

DG

h

ρ

i

F

E

X.

Conjecture

863

.

ρ

d

F

=

d

h

ρ

i

F

for a set

F

of reloids.

Proposition

864

.

ρ

id

RLD

(

A

)

= id

FCD

(

P

(

A

×

A

))

.

Proof.

D

ρ

id

RLD

(

A

)

E

x

= id

FCD

(

P

(

A

×

A

))

x

=

x

.

We can try to develop further theory by applying embedding of reloids into

funcoids for researching of properties of reloids.

Theorem

865

.

Reloid

f

is monovalued iff funcoid

ρf

is monovalued.

Proof.

ρf

is monovalued

(

ρf

)

(

ρf

)

1

v

1

Dst

ρf

ρ

(

f

f

1

)

v

1

Dst

ρf

ρ

(

f

f

1

)

v

id

FCD

(

P

(Dst

f

×

Dst

f

))

ρ

(

f

f

1

)

v

ρ

id

RLD

(Dst

f

)

f

f

1

v

id

RLD

(Dst

f

)

f

is monovalued

.