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18.4. IDENTITY STAROIDS AND MULTIFUNCOIDS

294

Remark

1525

.

v

on the diagram means inequality which can become strict

for some

A

and

n

.

18.4.7. Identity staroids represented as meets and joins.

Proposition

1526

.

id

Strd

a

[

n

]

=

d

n

Strd

id

A

[

n

]

A

a

o

for every filter

a

on a powerset

where the meet may be taken on every of the following posets: anchored relations,

staroids.

Proof.

That id

Strd

a

[

n

]

v↑

Strd

id

A

[

n

]

for every

A

a

is obvious.

Let

f

v↑

Strd

id

A

[

n

]

for every

A

a

.

L

GR

f

L

GR

Strd

id

A

[

n

]

⇒ ∀

A

a

:

A

l

i

n

L

i

6

A

A

l

i

n

L

i

6

a

L

GR id

Strd

a

[

n

]

.

Thus

f

v

id

Strd

a

[

n

]

.

Proposition

1527

.

ID

Strd

A

[

n

]

=

F

ID

Strd

a

[

n

]

a

atoms

A

=

F

n

a

n

Strd

a

atoms

A

o

where the join

may be taken on every of the following posets: anchored relations, staroids, com-

pletary staroids, provided that

A

is a filter on a set.

Proof.

ID

Strd

A

[

n

]

w

ID

Strd

a

[

n

]

for every

a

atoms

A

is obvious.

Let

f

w

ID

Strd

a

[

n

]

for every

a

atoms

A

. Then

L

GR ID

Strd

a

[

n

]

:

L

GR

f

that is

L

form

f

:

MEET

L

i

i

n

∪ {

a

}

L

GR

f

.

But

a

atoms

A

: MEET

L

i

i

n

∪ {

a

}

⇔ ∃

a

atoms

A

:

A

l

i

n

L

i

6

a

A

l

i

n

L

i

6 A ⇔

L

ID

Strd

A

[

n

]

.

So

L

ID

Strd

A

[

n

]

L

GR

f

. Thus

f

w

ID

Strd

A

[

n

]

.

Then use the fact that ID

Strd

a

[

n

]

=

a

n

Strd

.

Proposition

1528

.

id

Strd

A

[

n

]

=

F

id

Strd

a

[

n

]

a

atoms

A

where the meet may be taken on

every of the following posets: anchored relations, staroids, provided that

A

is a

filter on a set.

Proof.

id

Strd

A

[

n

]

w

id

Strd

a

[

n

]

for every

a

atoms

A

is obvious.

Let

f

w

id

Strd

a

[

n

]

for every

a

atoms

A

. Then

L

GR id

Strd

a

[

n

]

:

L

GR

f

that is

L

form

f

:

 

Z

l

i

n

L

i

6

a

L

GR

f

!

.

But

a

atoms

A

:

d

Z

i

n

L

i

6

a

d

Z

i

n

L

i

6 A ⇔

L

id

Strd

A

[

n

]

.

So

L

id

Strd

A

[

n

]

L

GR

f

. Thus

f

w

id

Strd

A

[

n

]

.