background image

14.1. ORDERING OF FILTERS

242

6

7

Let

f

be an injective reloid such that im

f

v A

and dom

f

w B

. Then

im

f

|

B

v A

and dom

f

|

B

=

B

. So

f

|

B

Hom

CoMonRld

v

,

=

(

A

,

B

).

2

5

,

3

6

,

4

7

By the lemma.

Theorem

1291

.

For every filters

A

and

B

the following are equivalent:

1

.

A ≥

2

B

.

2

. Hom

MonRld

=

,

=

(

A

,

B

)

6

=

.

3

. Hom

CoMonRld

=

,

=

(

A

,

B

)

6

=

.

Proof.

1

2

Let

A ≥

2

B

that is

B

=

FCD

f

A

for some

Set

-morphism

f

: Base(

A

)

Base(

B

). Then dom(

RLD

f

)

|

A

=

A

and

im(

RLD

f

)

|

A

= im(

FCD

)(

RLD

f

)

|

A

= im(

FCD

f

)

|

A

=

FCD

f

A

=

B

.

So (

RLD

f

)

|

A

is a sought for reloid.

2

1

There exists a monovalued reloid

f

with domain

A

such that

h

(

FCD

)

f

iA

=

B

. By corollary

1327

below, there exists a

Set

-morphism

F

: Base(

A

)

Base(

B

) such that

f

= (

RLD

F

)

|

A

. Thus

FCD

F

A

= im(

FCD

F

)

|

A

= im(

FCD

)(

RLD

F

)

|

A

= im(

FCD

)

f

= im

f

=

B

.

Thus

A ≥

2

B

is testified by the morphism

F

.

2

3

By the lemma.

Theorem

1292

.

The following are categories (with reloid composition):

1

.

MonRld

v

,

w

;

2

.

MonRld

v

,

=

;

3

.

MonRld

=

,

=

;

4

.

CoMonRld

v

,

w

;

5

.

CoMonRld

v

,

=

;

6

.

CoMonRld

=

,

=

.

Proof.

We will prove only the first three. The rest follow from duality. We

need to prove only that composition of morphisms is a morphism, because associa-

tivity and existence of identity morphism are evident. We have:

1

Let

f

Hom

MonRld

v

,

w

(

A

,

B

),

g

Hom

MonRld

v

,

w

(

B

,

C

). Then dom

f

v

A

, im

f

w B

, dom

g

v B

, im

g

w C

. So dom(

g

f

)

v A

, im(

g

f

)

w C

that is

g

f

Hom

MonRld

v

,

w

(

A

,

C

).

2

Let

f

Hom

MonRld

v

,

=

(

A

,

B

),

g

Hom

MonRld

v

,

=

(

B

,

C

). Then dom

f

v

A

, im

f

=

B

, dom

g

v B

, im

g

=

C

. So dom(

g

f

)

v A

, im(

g

f

) =

C

that is

g

f

Hom

MonRld

v

,

=

(

A

,

C

).

3

Let

f

Hom

MonRld

=

,

=

(

A

,

B

),

g

Hom

MonRld

=

,

=

(

B

,

C

). Then dom

f

=

A

, im

f

=

B

, dom

g

=

B

, im

g

=

C

. So dom(

g

f

) =

A

, im(

g

f

) =

C

that is

g

f

Hom

MonRld

=

,

=

(

A

,

C

).

Definition

1293

.

Let

BijRld

be the groupoid of all bijections of the category

of reloid triples. Its objects are filters and its morphisms from a filter

A

to filter

B

are monovalued injective reloids

f

such that dom

f

=

A

and im

f

=

B

.

Theorem

1294

.

Filters

A

and

B

are isomorphic iff Hom

BijRld

(

A

,

B

)

6

=

.

Proof.