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13.2. ORDERING OF FILTERS

182

Obvious

976

.

(

1

)

(

2

).

Definition

977

.

Let

Q

and

R

be binary relations on the set of filters. I

will denote

MonRld

Q,R

the directed multigraph with objects being filters and

morphisms such monovalued reloids

f

that (dom

f

)

Q

A

and (im

f

)

R

B

.

I will also denote

CoMonRld

Q,R

the directed multigraph with objects being

filters and morphisms such injective reloids

f

that (im

f

)

Q

A

and (dom

f

)

R

B

.

These are essentially the duals.

Some of these directed multigraphs are categories with reloid composition (see

below). By abuse of notation I will denote these categories the same as these

directed multigraphs.

Theorem

978

.

For every filters

A

and

B

the following are equivalent:

FiXme

:

Looks like an error. Check items 5–7 carefully.

1

.

A ≥

1

B

.

2

. Mor

MonRld

=

,

w

(

A

;

B

)

6

=

.

3

. Mor

MonRld

v

,

w

(

A

;

B

)

6

=

.

4

. Mor

MonRld

v

,

=

(

A

;

B

)

6

=

.

5

. Mor

CoMonRld

=

,

w

(

A

;

B

)

6

=

.

6

. Mor

CoMonRld

v

,

w

(

A

;

B

)

6

=

.

7

. Mor

CoMonRld

v

,

=

(

A

;

B

)

6

=

.

Proof.

1

2

There exists a

Set

-morphism

f

: Base(

A

)

Base(

B

) such that

B v

FCD

f

A

. We have

dom(

RLD

f

)

|

A

=

A u >

F

(Base(

A

))

=

A

and

im(

RLD

f

)

|

A

= im(

FCD

)(

RLD

f

)

|

A

= im(

FCD

f

)

|

A

=

FCD

f

A w B

.

Thus (

RLD

f

)

|

A

is a monovalued reloid such that dom(

RLD

f

)

|

A

=

A

and im(

RLD

f

)

|

A

w B

.

2

3

,

4

3

,

5

6

,

7

6

Obvious.

3

1

We have

B

v

h

(

FCD

)

f

iA

for a monovalued reloid

f

RLD

(Base(

A

); Base(

B

)).

Then there exists a

Set

-morphism

F

: Base(

A

)

Base(

B

) such that

B v

FCD

F

A

that is

A ≥

1

B

.

6

7

dom

f

|

B

=

B

and im

f

|

B

w A

.

2

5

,

3

6

,

4

7

By duality.

Theorem

979

.

For every filters

A

and

B

the following are equivalent:

1

.

A ≥

2

B

.

2

. Mor

MonRld

=

,

=

(

A

;

B

)

6

=

.

3

. Mor

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.