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10.5. EMBEDDING RELOIDS INTO FUNCOIDS

218

is defined by the formulas:

h

ρf

i

x

=

f

x

and

ρf

1

y

=

f

1

y

where

x

are endoreloids on Src

f

and

y

are endoreloids on Dst

f

.

Proposition

1165

.

It is really a funcoid (if we equate reloids

x

and

y

with

corresponding filters on Cartesian products of sets).

Proof.

y

6 h

ρf

i

x

y

6

f

x

f

1

y

6

x

ρf

1

y

6

x

.

Corollary

1166

.

(

ρf

)

1

=

ρf

1

.

Definition

1167

.

It can be continued to arbitrary funcoids

x

having destina-

tion Src

f

by the formula

h

ρ

f

i

x

=

h

ρf

i

id

Src

f

x

=

f

x

.

Proposition

1168

.

ρ

is an injection.

Proof.

Consider

x

= id

Src

f

.

Proposition

1169

.

ρ

(

g

f

) = (

ρg

)

(

ρf

).

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

1170

.

ρ

d

F

=

d

h

ρ

i

F

for a set

F

of reloids.

Proof.

It’s enough to prove

h

ρ

d

F

i

X

=

d

h

ρ

i

F

X

for a set

X

.

Really,

D

ρ

l

F

E

X

=

D

ρ

l

F

E

X

=

l

F

◦ ↑

X

=

l

f

◦ ↑

X

f

F

=

l

h

ρf

i ↑

X

f

F

=

l

ρf

f

F

X

=

D

l

h

ρ

i

F

E

X.

Conjecture

1171

.

ρ

d

F

=

d

h

ρ

i

F

for a set

F

of reloids.

Proposition

1172

.

ρ

1

RLD

A

= 1

FCD

P

(

A

×

A

)

.

Proof.

ρ

1

RLD

A

x

= 1

RLD

A

x

=

x

=

D

1

FCD

P

(

A

×

A

)

E

x

.

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

funcoids for researching of properties of reloids.

Theorem

1173

.

Reloid

f

is monovalued iff funcoid

ρf

is monovalued.