background image

8

Y

2

up

X

:

d

i

2

n

Z

Y

i

u "

A

=

/ 0

does not hold for

n

=

N

,

X

i

=

"

(

¡

1/

i

; 0)

for

i

2

n

,

A

= (

¡1

; 0)

.

To show this, it's enough to prove

d

i

2

n

Z

Y

i

u "

A

=

/ 0

for

Y

i

=

"

(

¡

1 /

i

; 0)

but this is obvious since

d

i

2

n

Z

Y

i

= 0

.

On the other hand,

d

i

2

n

A

X

i

u "

A

=

/ 0

for the same

X

and

A

.

The above theorems are summarized in the following diagram:

"

Strd

id

A

[

n

]

=

id

"

A

[

n

]

Strd

"

Strd

id

A

[

n

]

=

id

"

A

[

n

]

Strd

w

ID

"

A

[

n

]

Strd

w

ID

"

A

[

n

]

Strd

Remark 18.73.

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 18.74.

id

a

[

n

]

Strd

=

d

f"

Strd

id

A

[

n

]

j

A

2

a

g

for every lter

a

on a powerset where the

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

Proof.

That id

a

[

n

]

Strd

v "

Strd

id

A

[

n

]

for every

A

2

a

is obvious.

Let

f

v "

Strd

id

A

[

n

]

for every

A

2

a

.

L

2

GR

f

)

L

2

GR

"

Strd

id

A

[

n

]

) 8

A

2

a

:

d

i

2

n

A

L

i

/

A

)

d

i

2

n

A

L

i

/

a

)

L

2

GR id

a

[

n

]

Strd

. Thus

f

v

id

a

[

n

]

Strd

.

Proposition 18.75.

ID

A

[

n

]

Strd

=

ID

a

[

n

]

Strd

j

a

2

atoms

A

 

=

F

f

a

Strd

n

j

a

2

atoms

Ag

where the join

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

A

is a lter on a set.

Proof.

ID

A

[

n

]

Strd

w

ID

a

[

n

]

Strd

for every

a

2

atoms

A

is obvious.

Let

f

w

ID

a

[

n

]

Strd

for every

a

2

atoms

A

. Then

8

L

2

GR ID

a

[

n

]

Strd

:

L

2

GR

f

that is

8

L

2

form

f

: (

MEET

(

f

L

i

j

i

2

n

g [ f

a

g

)

)

L

2

GR

f

)

:

But

9

a

2

atoms

A

:

MEET

(

f

L

i

j

i

2

n

g [ f

a

g

)

, 9

a

2

atoms

A

:

d

i

2

n

A

L

i

/

a

(

d

i

2

n

A

L

i

/

A ,

L

2

ID

A

[

n

]

Strd

.

So

L

2

ID

A

[

n

]

Strd

)

L

2

GR

f

. Thus

f

w

ID

A

[

n

]

Strd

.

Then use the fact that ID

a

[

n

]

Strd

=

a

Strd

n

.

Proposition 18.76.

id

A

[

n

]

Strd

=

id

a

[

n

]

Strd

j

a

2

atoms

A

 

where the meet may be taken on every of

the following posets: anchored relations, staroids, provided that

A

is a lter on a set.

Proof.

id

A

[

n

]

Strd

w

id

a

[

n

]

Strd

for every

a

2

atoms

A

is obvious.

Let

f

w

id

a

[

n

]

Strd

for every

a

2

atoms

A

. Then

8

L

2

GR id

a

[

n

]

Strd

:

L

2

GR

f

that is

8

L

2

form

f

:

 

l

i

2

n

Z

L

i

/

a

)

L

2

GR

f

!

:

But

9

a

2

atoms

A

:

d

i

2

n

Z

L

i

/

a

(

d

i

2

n

Z

L

i

/

A ,

L

2

id

A

[

n

]

Strd

.

250

Identity staroids