background image

21.11. MULTIRELOIDS

348

2

.

StarComp(

m, λi

arity

m

: 1

Obj

m

i

) =

RLD

(

A

)

l

(

StarComp(

A, H

)

A

GR

m, H

Q

i

arity

m

GR 1

Obj

m

i

)

=

RLD

(

A

)

l

(

StarComp(

A, λi

arity

m

:

H

i

)

A

GR

m, H

Q

i

arity

m

GR 1

Obj

m

i

)

=

RLD

(

A

)

l

(

StarComp(

A, λi

arity

m

: 1

X

i

)

A

GR

m, X

Q

i

arity

m

Obj

m

i

)

=

RLD

(

A

)

l

(

(

A

Q

X

)

A

GR

m, X

Q

i

arity

m

Obj

m

i

)

=

RLD

(

A

)

l

A

A

GR

m

=

m.

3

Using properties of generalized filter bases,

b

6

StarComp(

a, f

)

A

GR

a, B

GR

b, F

Y

i

n

GR

f

i

:

B

6

StarComp(

A, F

)

A

GR

a, B

GR

b, F

Y

i

n

GR

f

i

:

B

6

*

(

C

)

Y

F

+

A

A

GR

a, B

GR

b, F

Y

i

n

GR

f

i

:

A

6

*

(

C

)

Y

F

1

+

B

A

GR

a, B

GR

b, F

Y

i

n

GR

f

i

:

A

6

StarComp(

B, F

)

a

6

StarComp(

b, f

)

.

Definition

1762

.

Let

f

be a multireloid of the form

A

. Then for

i

dom

A

RLD

Pr

i

f

=

F

l

D

Pr

i

E

GR

f.

Proposition

1763

.

up Pr

RLD

i

f

=

h

Pr

i

i

GR

f

for every multireloid

f

and

i

arity

f

.

Proof.

It’s enough to show that

h

Pr

i

i

GR

f

is a filter.

That

h

Pr

i

i

GR

f

is an upper set is obvious.

Let

X, Y

∈ h

Pr

i

i

GR

f

. Then there exist

F, G

GR

f

such that

X

= Pr

i

F

,

Y

= Pr

i

G

. Then

X

Y

Pr

i

(

F

G

)

∈ h

Pr

i

i

GR

f

. Thus

X

Y

∈ h

Pr

i

i

GR

f

.

Definition

1764

.

Q

RLD

X

=

d

RLD

(

λi

dom

X

:Base(

X

i

))

X

up

Q

X

Q

X

for every indexed

family

X

of filters on powersets.

Proposition

1765

.

Pr

RLD

k

Q

RLD

x

=

x

k

for every indexed family

x

of proper

filters.

Proof.

up Pr

RLD

k

Q

RLD

x

=

h

Pr

k

i

Q

RLD

x

= up

x

k

.