background image

b

StarComp

(

a

;

f

)

⇔ ∀

A

a, B

B , F

Q

i

n

f

i

:

B

StarComp

(

A

;

F

)

⇔ ∀

A

a, B

B ,

F

Q

i

n

f

i

:

B

D

Q

(

C

)

F

E

A

⇔ ∀

A

a, B

B , F

Q

i

n

f

i

:

A

D

Q

(

C

)

F

1

E

B

A

a, B

B , F

Q

i

n

f

i

:

A

StarComp

(

B

;

F

)

a

StarComp

(

b

;

f

)

.

Definition 158.

Let

f

is a multireloid of the form

A

. Then for

i

dom

A

Pr

i

RLD

f

=

l

h↑

A

i

ih

Pr

i

i

f .

Definition 159.

Q

RLD

X

=

d

RLD

(

λi

dom

X

:

Base

(

X

i

))

Q

X

|

X

∈ X

 

for every indexed family

X

of filters on powersets.

Proposition 160.

Pr

k

RLD

Q

RLD

x

=

x

k

for every indexed family

x

of proper filters.

Proof.

It follows from

h

Pr

k

i

RLD

(

λi

dom

X

:

Base

(

X

i

))

Q

X

|

X

x

 

=

d

{

X

|

X

x

}

=

x

.

Conjecture 161.

GR StarComp

(

a

;

λi

n

:

f

i

g

i

) =

GR StarComp

(

a

;

f

)

GR StarComp

(

a

;

g

)

for

a multireloid

a

and indexed families

f

and

g

of multireloids where Src

f

i

=

Src

g

i

and Dst

f

i

=

Dst

g

i

.

Conjecture 162.

GR StarComp

(

a

b

;

f

) =

GR StarComp

(

a

;

f

)

GR StarComp

(

b

;

f

)

if

f

is a

reloid and

a

,

b

are multireloids of the same form, composable with

f

.

Theorem 163.

Q

RLD

A

=

F  Q

RLD

a

|

a

Q

i

dom

A

atoms

A

i

 

for every indexed family

A

of

filters on powersets.

Proof.

Obviously

Q

RLD

A

F  Q

RLD

a

|

a

Q

i

dom

A

atoms

A

i

 

.

Reversely, let

K

F  Q

RLD

a

|

a

Q

i

dom

A

atoms

A

i

 

. Then for every

i

dom

A

we have

K

Q

RLD

a

i

for every

a

i

Q

j

dom

A

atoms

A

j

and so

K

Q

X

i

for some

X

i

Q

j

dom

A

A

j

.

Consequently

K

F

i

dom

A

Q

X

i

=

F

i

dom

A

Q

j

dom

A

X

i, j

=

Q

j

dom

A

F

i

dom

A

X

i,j

Q

j

dom

A

Z

j

for some

Z

j

A

j

. So

K

Q

RLD

A

.

Theorem 164.

Let

a

,

b

be indexed families of filters on powersets of the same form

A

. Then

Y

RLD

a

Y

RLD

b

=

Y

i

dom

A

RLD

(

a

i

b

i

)

.

Proof.

Y

RLD

a

Y

RLD

b

=

(

RLD

(

A

)

(

P

Q

)

|

P

Y

RLD

a, Q

Y

RLD

b

)

=

RLD

(

A

)

Y

p

Y

q

|

p

Y

a, q

Y

b

 

=

RLD

(

A

)

 

Y

i

dom

A

(

p

i

q

i

)

!

|

p

Y

a, q

Y

b

=

(

RLD

(

A

)

Y

r

|

r

Y

i

dom

A

(

a

i

b

i

)

)

=

Y

i

dom

A

RLD

(

a

i

b

i

)

.

Theorem 165.

If

S

P

Q

i

dom

A

F

(

A

i

)

where

A

is an indexed family of sets, then

l

(

Y

RLD

a

|

a

S

)

=

Y

i

dom

A

RLD

l

F

(

A

i

)

Pr

i

S.

26

Section 13