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Proposition 17.168.

Pr

i

RLD

f

=

h

Pr

i

i

GR

a

for every multireloid

a

and

i

2

arity

a

, given a set

A

 h

Pr

i

i

a

.

[TODO: Describe it with anchored relations instead.]

Proof.

It's enough to show that

h

Pr

i

i

GR

f

is a lter.

That

h

Pr

i

i

GR

f

is an upper set is obvious.

Let

X ; Y

2 h

Pr

i

i

GR

f

. Then there exist

F ; G

2

GR

f

such that

X

=

Pr

i

F

,

Y

=

Pr

i

G

. Then

X

\

Y

Pr

i

(

F

\

G

)

2 h

Pr

i

i

GR

f

. Thus

X

\

Y

2 h

Pr

i

i

GR

f

.

Denition 17.169.

Q

RLD

X

=

d

"

RLD

(

i

2

dom

X

:

Base

(

X

i

))

Q

X

j

X

2

Q

X

 

for every indexed

family

X

of lters on powersets.

Proposition 17.170.

Pr

k

RLD

Q

RLD

x

=

x

k

for every indexed family

x

of proper lters.

Proof.

Pr

k

RLD

Q

RLD

x

=

h

Pr

k

i

Q

RLD

x

=

x

k

.

Conjecture 17.171.

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 17.172.

Q

RLD

A

=

F Q

RLD

a

j

a

2

Q

i

2

dom

A

atoms

A

i

 

for every indexed family

A

of lters on powersets.

Proof.

Obviously

Q

RLD

A

w

F Q

RLD

a

j

a

2

Q

i

2

dom

A

atoms

A

i

 

.

Reversely, let

K

2

GR

F Q

RLD

a

j

a

2

Q

i

2

dom

A

atoms

A

i

 

.

Consequently

K

2

GR

Q

RLD

a

for every

a

2

Q

i

2

dom

A

atoms

A

i

;

K

Q

X

and thus

K

S

X

2

Q

a

Q

X

for every

X

2

Q

a

.

But

S

X

2

Q

a

Q

X

=

Q

i

2

dom

A

S

h

Pr

i

i

X

Q

j

2

dom

A

Z

j

for some

Z

j

2

A

j

because

h

Pr

i

i

X

2

a

i

and our lattice is atomistic. So

K

2

GR

Q

RLD

A

.

Theorem 17.173.

Let

a

,

b

be indexed families of lters on powersets of the same form

A

. Then

Y

RLD

a

u

Y

RLD

b

=

Y

i

2

dom

A

RLD

(

a

i

u

b

i

)

:

Proof.

Y

RLD

a

u

Y

RLD

b

=

l

(

"

RLD

(

A

)

(

P

\

Q

)

j

P

2

GR

Y

RLD

a; Q

2

GR

Y

RLD

b

)

=

l

"

RLD

(

A

)

¡Y

p

\

Y

q

j

p

2

Y

a; q

2

Y

b

 

=

l

8

<

:

"

RLD

(

A

)

  Y

i

2

dom

A

(

p

i

\

q

i

)

!

j

p

2

Y

a; q

2

Y

b

9

=

;

=

l

(

"

RLD

(

A

)

Y

r

j

r

2

Y

i

2

dom

A

(

a

i

u

b

i

)

)

=

Y

i

2

dom

A

RLD

(

a

i

u

b

i

)

:

Theorem 17.174.

If

S

2

P

Q

i

2

dom

A

F

(

A

i

)

where

A

is an indexed family of sets, then

l

(

Y

RLD

a

j

a

2

S

)

=

Y

i

2

dom

A

RLD

l

"

F

(

A

i

)

Pr

i

S:

228

Multifuncoids and staroids