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8. MORE RESULTS ON RESTRICTED IDENTITIES

21

1

.

S

from a Hom-set

RLD

(

A, B

) to End

RLD

(small sets) is an order embed-

ding.

2

.

S

from the category

RLD

to End

RLD

(small sets) is a prefunctor.

3

.

S

from unfixed reloids is an order embedding and a prefunctor (= semi-

group homomorphism).

Proof.

1

That it’s monotone is obvious. That it is an injection follows from

S

for

filters being an injection.

2

Let

f

and

g

be composable reloids.

If

H

up

S

(

g

f

) then

H

H

0

up(

g

f

),

H

0

G

F

for some

H

0

,

F

up

f

and

G

up

g

. Consequently

F

GR

S

f

,

G

GR

S

g

. So

G

F

up(

S

g

S

f

)

and thus

S

(

g

f

)

w

S

g

S

f

.

Whenever

H

up(

S

g

S

f

), we have

H

G

F

where

F

up

S

f

,

G

up

S

g

. Thus

F

F

0

up

f

,

G

G

0

up

g

;

H

G

0

F

0

up(

g

f

) for some

F

0

,

G

0

and so

H

up(

S

(

g

f

)). So

S

g

S

f

w

S

(

g

f

).

So

S

(

g

f

) =

S

g

S

f

.

3

That it is a prefunctor easily follows from the above.

Suppose

f

,

g

are unfixed reloids and

S

f

=

S

g

. Let

F

f

,

G

g

and thus

S

F

=

S

G

. It is enough to prove that

F

G

.

Really,

S

F

=

S

G

S

GR

F

=

S

GR

G

GR

F

GR

G

GR

G

=

(GR

F

)

÷

(dom

G

×

im

G

)

G

=

F

÷

(dom

G

×

im

G

) =

ι

dom

G,

im

G

F

. Similarly

F

=

ι

dom

F,

im

F

G

. So

F

G

.

I yet failed to generalize propositions

2137

and

2138

The generalization may

require first research pointfree reloids.

8. More results on restricted identities

In the next three propositions assume

A

Z

,

A

3

X

v

A

.

Proposition

2148

.

id

Rel

(

A

)

X

= id

Rel

(

A,A

)

[

X

]

.

Proof.

id

Rel

(

A,A

)

[

X

]

= id

Rel

(

A,A

)

X

= id

Rel

(

A

)

X

.

Proposition

2149

.

id

FCD

(

A

)

X

= id

FCD

(

A,A

)

[

X

]

.

Proof.

D

id

FCD

(

A,A

)

[

X

]

E

X

= ([

X

]

u

[

X

])

÷

A

= [

X u

X

])

÷

A

=

X u

X

=

D

id

FCD

(

A

)

X

E

X

for

A

3 X v

A

.

Proposition

2150

.

id

RLD

(

A

)

X

= id

RLD

(

A,A

)

[

X

]

.

Proof.

id

RLD

(

A,A

)

[

X

]

= id

RLD

[

X

]

÷

(

A

A

)

÷

(

A

×

A

) = id

RLD

X

÷

(

A

×

A

) = id

RLD

(

A

)

X

.

As a generalization of three last propositions, define for every category

C

with

restricted identities:

Definition

2151

.

id

C

(

A

)

X

= id

C

(

A,A

)

[

X

]

.

Proposition

2152

.

n

(

A,

Au

A

)

A∈

F

(

U

)

o

is a function and moreover is an order iso-

morphism for a set

A

U

.