background image

Strd

id

A

[

n

]

=

id

A

[

n

]

Strd

Strd

id

A

[

n

]

=

id

A

[

n

]

Strd

ID

A

[

n

]

Strd

ID

A

[

n

]

Strd

Remark 73.

on the diagram means inequality which can become strict for some

A

and

n

.

6.7 Identity staroids represented as meets and joins

Proposition 74.

id

a

[

n

]

Strd

=

d

{↑

Strd

id

A

[

n

]

|

A

a

}

for every set-theoretic filter

a

where the meet

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

Proof.

That id

a

[

n

]

Strd

⊑ ↑

Strd

id

A

[

n

]

for every

A

a

is obvious.

Let

f

⊑ ↑

Strd

id

A

[

n

]

for every

A

a

.

L

GR

f

L

GR

Strd

id

A

[

n

]

⇒ ∀

A

a

:

d

i

n

A

L

i

A

d

i

n

A

L

i

a

L

GR id

a

[

n

]

Strd

. Thus

f

id

a

[

n

]

Strd

.

Proposition 75.

ID

A

[

n

]

Strd

=

ID

a

[

n

]

Strd

|

a

atoms

A

 

=

F

{

a

Strd

n

|

a

atoms

A}

where the meet

may be taken on every of the following posets: anchored relations, staroids, completary staroids,
provided that

A

is a filter on a set.

Proof.

ID

A

[

n

]

Strd

ID

a

[

n

]

Strd

for every

a

atoms

A

is obvious.

Let

f

ID

a

[

n

]

Strd

for every

a

atoms

A

. Then

L

GR ID

a

[

n

]

Strd

:

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

:

d

i

n

A

L

i

a

d

i

n

A

L

i

A ⇔

L

ID

A

[

n

]

Strd

.

So

L

ID

A

[

n

]

Strd

L

GR

f

. Thus

f

ID

A

[

n

]

Strd

.

Then use the fact that ID

a

[

n

]

Strd

=

a

Strd

n

.

Proposition 76.

id

A

[

n

]

Strd

=

id

a

[

n

]

Strd

|

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

A

[

n

]

Strd

id

a

[

n

]

Strd

for every

a

atoms

A

is obvious.

Let

f

id

a

[

n

]

Strd

for every

a

atoms

A

. Then

L

GR id

a

[

n

]

Strd

:

L

GR

f

that is

L

form

f

:

 

l

i

n

Z

L

i

a

L

GR

f

!

.

But

a

atoms

A

:

d

i

n

Z

L

i

a

d

i

n

Z

L

i

A ⇔

L

id

A

[

n

]

Strd

.

So

L

id

A

[

n

]

Strd

L

GR

f

. Thus

f

id

A

[

n

]

Strd

.

7 Finite case

Theorem 77.

Let

n

be a finite set.

1. id

A

[

n

]

Strd

=

ID

A

[

n

]

Strd

if

A

and

Z

are meet-semilattices and

(

A

;

Z

)

is a finitely meet-closed filtrator.

10

Section 7