17.12. MULTIRELOIDS

265

Conjecture

1368

.

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

1369

.

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

w

F

Q

RLD

a

a

Q

i

dom

A

atoms

A

i

.

Reversely, let

K

GR

F

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

X

Q

a

Q

X

for every

X

Q

a

.

But

S

X

Q

a

Q

X

=

Q

i

dom

A

S

h

Pr

i

i

X

Q

j

dom

A

Z

j

for some

Z

j

A

j

because

h

Pr

i

i

X

a

i

and our lattice is atomistic. So

K

GR

Q

RLD

A

.

Theorem

1370

.

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.

RLD

Y

a

u

RLD

Y

b

=

l

(

RLD

(

A

)

(

P

Q

)

P

GR

Q

RLD

a, Q

Q

RLD

b

)

=

l

RLD

(

A

)

(

Q

p

Q

q

)

p

Q

a, q

Q

b

=

l

(

RLD

(

A

)

Q

i

dom

A

(

p

i

q

i

)

p

Q

a, q

Q

b

)

=

l

RLD

(

A

)

Q

r

r

Q

i

dom

A

(

a

i

u

b

i

)

=

RLD

Y

i

dom

A

(

a

i

u

b

i

)

.

Theorem

1371

.

If

S

P

Q

i

dom

A

F

(

A

i

) where

A

is an indexed family of sets,

then

l

(

Q

RLD

a

a

S

)

=

RLD

Y

i

dom

A

l

D

F

(

A

i

)

E

Pr

i

S.