background image

21.8. ON PRODUCTS OF STAROIDS

332

A

i

6

L

i

. Then, by properties of generalized filter bases,

i

dom

A

:

a

i

6

L

i

that

is

L

GR

Q

Strd

(

F

)

a

. So

f

v

Q

Strd

(

F

)

a

and thus

Q

Strd

(

F

)

a

is the greatest lower

bound of

 Q

Strd

(

F

)

A

A

S

.

Conjecture

1696

.

Let

F

be a family of sets of filters on meet-semilattices

with least elements. Let

a

Q

F

,

S

P

Q

F

be a generalized filter base,

d

S

=

a

,

f

is a staroid of the form

Q

F

. Then

Strd

(

F

)

Y

a

6

f

⇔ ∀

A

S

:

Strd

(

Z

)

Y

A

6

f.

21.8. On products of staroids

Definition

1697

.

Q

(

D

)

F

=

uncurry

z

z

Q

F

(

reindexation product

) for every in-

dexed family

F

of relations.

Definition

1698

.

Reindexation product

of an indexed family

F

of anchored

relations is defined by the formulas:

form

(

D

)

Y

F

= uncurry(form

F

) and GR

(

D

)

Y

F

=

(

D

)

Y

(GR

F

)

.

Obvious

1699

.

1

. form

Q

(

D

)

F

=

n

((

i,j

)

,

(form

F

i

)

j

)

i

dom

F,j

arity

F

i

o

;

2

. GR

Q

(

D

)

F

=

 

((

i,j

)

,

(

zi

)

j

)

i

dom

F,j

arity

Fi

 

z

Q

(GR

F

)

.

Proposition

1700

.

Q

(

D

)

F

is an anchored relation if every

F

i

is an anchored

relation.

Proof.

We need to prove GR

Q

(

D

)

F

P

Q

form

Q

(

D

)

F

that is

GR

Q

(

D

)

F

Q

form

Q

(

D

)

F

;

 

((

i,j

)

,

(

zi

)

j

)

i

dom

F,j

arity

Fi

 

z

Q

(GR

F

)

Q

n

((

i,j

)

,

(form

F

i

)

j

)

i

dom

F,j

arity

F

i

o

;

z

Q

(GR

F

)

, i

dom

F, j

arity

F

i

: (

zi

)

j

(form

F

i

)

j

.

Really,

zi

GR

F

i

Q

(form

F

i

) and thus (

zi

)

j

(form

F

i

)

j

.

Obvious

1701

.

arity

Q

(

D

)

F

=

`

i

dom

F

arity

F

i

=

n

(

i,j

)

i

dom

F,j

arity

F

i

o

.

Definition

1702

.

f

×

(

D

)

g

=

Q

(

D

)

J

f, g

K

.

Lemma

1703

.

Q

(

D

)

F

is an upper set if every

F

i

is an upper set.

Proof.

We need to prove that

Q

(

D

)

F

is an upper set. Let

a

Q

(

D

)

F

and

an anchored relation

b

w

a

of the same form as

a

. We have

a

= uncurry

z

for some

z

Q

F

that is

a

(

i, j

) = (

zi

)

j

for all

i

dom

F

and

j

dom

F

i

where

zi

F

i

.

Also

b

(

i, j

)

w

a

(

i, j

). Thus (curry

b

)

i

w

zi

; curry

b

Q

F

because every

F

i

is an

upper set and so

b

Q

(

D

)

F

.

Proposition

1704

.

Let

F

be an indexed family of anchored relations and

every (form

F

)

i

be a join-semilattice.

1

.

Q

(

D

)

F

is a prestaroid if every

F

i

is a prestaroid.

2

.

Q

(

D

)

F

is a staroid if every

F

i

is a staroid.

3

.

Q

(

D

)

F

is a completary staroid if every

F

i

is a completary staroid.