background image

21.11. MULTIRELOIDS

349

Conjecture

1766

.

GR StarComp(

a

t

b, f

) = GR StarComp(

a, f

)

t

GR StarComp(

b, f

) if

f

is a reloid and

a

,

b

are multireloids of the same form,

composable with

f

.

Theorem

1767

.

Q

RLD

A

=

d

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

w

d

Q

RLD

a

a

Q

i

dom

A

atoms

A

i

.

Reversely, let

K

GR

d

Q

RLD

a

a

Q

i

dom

A

atoms

A

i

.

Consequently

K

GR

Q

RLD

a

for every

a

Q

i

dom

A

atoms

A

i

;

K

Q

X

and

thus

K

S

a

Q

i

dom

a

atoms

A

i

Q

X

a

for some

X

a

Q

i

dom

a

atoms

A

i

.

But

S

a

Q

i

dom

a

atoms

A

i

Q

X

a

=

Q

i

dom

A

S

a

atoms

A

i

h

Pr

i

i

X

a

Q

j

dom

A

Z

j

for some

Z

j

up

A

j

because

h

Pr

i

i

X

up

a

i

and our lattice

is atomistic. So

K

GR

Q

RLD

A

.

Theorem

1768

.

Let

a

,

b

be indexed families of filters on powersets of the same

form

A

. Then

RLD

Y

a

u

RLD

Y

b

=

RLD

Y

i

dom

A

(

a

i

u

b

i

)

.

Proof.

up

 

RLD

Y

a

u

RLD

Y

b

!

=

RLD

(

A

)

l

(

P

Q

P

GR

Q

RLD

a, Q

Q

RLD

b

)

=

RLD

(

A

)

l

Q

p

Q

q

p

up

Q

a, q

up

Q

b

=

RLD

(

A

)

l

Q

i

dom

A

(

p

i

q

i

)

p

Q

up

a, q

Q

up

b

=

RLD

(

A

)

l

Q

r

r

up

Q

i

dom

A

(

a

i

u

b

i

)

=

up

RLD

Y

i

dom

A

(

a

i

u

b

i

)

.

Theorem

1769

.

If

S

P

Q

i

dom

Z

F

(

Z

i

) where

Z

is an indexed family of

sets, then

l

a

S

RLD

Y

a

=

RLD

Y

i

dom

Z

F

(

Z

i

)

l

Pr

i

S.

Proof.

If

S

=

then

d

a

S

Q

RLD

a

=

d

=

>

RLD

(

Z

)

and

RLD

Y

i

dom

Z

F

(

Z

i

)

l

Pr

i

S

=

RLD

Y

i

dom

Z

F

(

Z

i

)

l

=

RLD

Y

i

dom

Z

>

F

(

Z

i

)

=

>

RLD

(

Z

)

,