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7. EXAMPLES OF CATEGORIES WITH RESTRICTED IDENTITIES

19

1

.

(

h

S

f

iX

)

÷

Dst

f

=

h

f

iX

. Thus for different

f

we have different

X 7→

h

S

f

iX

. So it is an injection. That it is a monotone function is obvious.

2

.

h

S

g

S

f

iX

=

h

S

g

ih

S

f

iX

=

h

S

g

i

[

h

f

i

(

X ÷

Src

f

)] = [

h

g

i

([

h

f

i

(

X ÷

Src

f

)]

÷

Src

g

)] = [

h

g

i

(

h

f

i

(

X ÷

Src

f

)

÷

Src

g

)] = [

h

g

ih

f

i

(

X ÷

Src

f

)] = [

h

g

f

i

(

X ÷

Src

f

)] =

h

S

(

g

f

)

iX

for every composable funcoids

f

and

g

and an unfixed fil-

ter

X

. Thus

S

g

S

f

=

S

(

g

f

).

3

.

To prove that it is an order embedding, it is enough to show that

f

g

implies

S

f

6

=

S

g

(monotonicity is obvious).

Let

f

g

that is

ι

A

0

t

A

1

,B

0

t

B

1

f

6

=

ι

A

0

t

A

1

,B

0

t

B

1

g

. Then there exist filter

X ∈

F

(

A

0

t

A

1

) such

that

h

ι

A

0

t

A

1

,B

0

t

B

1

f

iX 6

=

h

ι

A

0

t

A

1

,B

0

t

B

1

g

iX

.

Consequently,

h

S

f

iX

=

h

S

ι

A

0

t

A

1

,B

0

t

B

1

f

iX 6

=

h

S

ι

A

0

t

A

1

,B

0

t

B

1

g

iX

=

h

S

g

iX

.

It remains to prove that

S

G

S

F

=

S

(

G

F

) but it is equivalent to

S

g

S

f

=

S

(

g

f

) for arbitrarily taken

f

F

and

g

G

, what is already proved above.

Lemma

2136

.

For every meet-semilattice

a

6

b

and

c

w

b

implies

a

u

c

6

b

.

Proof.

Suppose

a

6

b

. Then there is a non-least

x

such that

x

v

a, b

. Thus

x

v

c

, so

x

v

a

u

c

. We have

a

u

c

6

b

.

FiXme

: Since here also for reloids.

Proposition

2137

.

S

(

X

×

Y

) =

X

×

pFCD

(

F

(

f

))

Y

for every unfixed filters

X

and

Y

.

Proof.

S

(

X

×

Y

) =

S

(

X

×

A,B

Y

) for arbitrary filters

A

, and

B

such that

X

v

[

A

] and

Y

v

[

B

]. So for every unfixed filter

X

we have

h

S

(

X

×

Y

)

iX

=

h

S

(

X

×

A,B

Y

)

iX

= [

h

X

×

A,B

Y

i

(

X ÷

A

)] = [

(

X

÷

A

)

×

FCD

(

Y

÷

B

)

(

X ÷

A

)]

Thus if

P 6

X

then (by the lemma)

P u

A

6

X

;

P ÷

A

6

X

÷

A

;

h

S

(

X

×

Y

)

iX

= [

Y

÷

B

] =

Y

.

if

X

then

P u

A

X

;

P ÷

A

X

÷

A

;

h

S

(

X

×

Y

)

iX

= [

] =

.

So

S

(

X

×

Y

) =

X

×

pFCD

(

F

(

f

))

Y

.

Proposition

2138

.

S

id

X

= id

pFCD

(

F

(

f

))

X

for every unfixed filter

X

.

Proof.

For every unfixed filter

X

we for arbitrary filters

A

and

B

such

that

X

v

[

A

]

u

[

B

] have

h

S

id

X

iX

=

D

S

[id

C

(

A,B

)

X

]

E

X

=

D

S

id

C

(

A,B

)

X

E

X

=

hD

id

C

(

A,B

)

X

E

(

X ÷

A

)

i

= [([

X ÷

A

]

u

X

)

÷

B

] = [(

X u

X

)

÷

B

] =

X u

X

.

Thus

S

id

X

= id

pFCD

(

F

(

f

))

X

.

7.3. Category

RLD

.

Definition

2139

.

f

÷

D

= (

A, B,

(GR

f

)

÷

D

) for every reloid

f

and a binary

relation

D

.

Category

RLD

can be considered as a category with restricted identities with

Z

being the set of all small sets,

A

is the set of unfixed filters, projection being the

projection function for the equivalence classes of filters, restricted identity being
defined by the formula

id

RLD

(

A,B

)

F

= id

RLD

F ÷

(

A

B

)

÷

(

A

×

B

)

.

We need to prove that the restricted identities conform to the axioms:

Proof.

The first five

axioms

are obvious. Let’s prove the remaining ones:

id

RLD

(

A,A

)

[

A

]

= id

RLD

[

A

]

÷

A

÷

(

A

×

A

) = id

RLD

A

÷

(

A

×

A

) = 1

RLD

A

.