21.19. IDENTITY STAROIDS AND MULTIFUNCOIDS

374

Question

1894

.

Is ID

Strd

A

[

n

]

principal for every principal filter

A

on a set and

index set

n

?

Proposition

1895

.

Strd

id

A

[

n

]

v

ID

Strd

A

[

n

]

for every set

A

.

Proof.

L

GR

Strd

id

A

[

n

]

L

GR id

Strd

A

[

n

]

⇔↑

A

6

Z

l

i

n

L

i

A

6

A

l

i

n

L

i

L

GR ID

Strd

A

[

n

]

.

Proposition

1896

.

Strd

id

A

[

n

]

@

ID

Strd

A

[

n

]

for some set

A

and index set

n

.

Proof.

L

GR

Strd

id

A

[

n

]

d

Z

i

n

L

i

6↑

A

what is not implied by

d

A

i

n

L

i

6↑

A

that is

L

GR ID

Strd

A

[

n

]

. (For a counter example take

n

=

N

,

L

i

=]0; 1

/i

[,

A

=

R

.)

Proposition

1897

.

Strd

id

A

[

n

]

=

id

Strd

A

[

n

]

.

Proof.

Strd

id

A

[

n

]

=

id

Strd

A

[

n

]

is obvious from the above.

Proposition

1898

.

Strd

id

A

[

n

]

v

ID

Strd

A

[

n

]

.

Proof.

X ∈

GR

Strd

id

A

[

n

]

up

X ⊆

GR

Strd

id

A

[

n

]

Y

up

X

:

Y

GR

Strd

id

A

[

n

]

⇔ ∀

Y

up

X

:

Y

GR id

Strd

A

[

n

]

Y

up

X

:

Z

l

i

n

Y

i

u ↑

A

6

=

⊥ ⇒

A

l

i

n

X

i

u ↑

A

6

=

⊥ ⇔ X ∈

GR ID

Strd

A

[

n

]

.

Proposition

1899

.

Strd

id

A

[

n

]

@

ID

Strd

A

[

n

]

for some set

A

.

Proof.

We need to prove

Strd

id

A

[

n

]

6

= ID

Strd

A

[

n

]

that is it’s enough to prove

(see the above proof) that

Y

up

X

:

d

Z

i

n

Y

i

u ↑

A

6

=

:

d

A

i

n

X

i

u ↑

A

6

=

.

A counter-example follows:

Y

up

X

:

d

Z

i

n

Y

i

u ↑

A

6

=

does not hold for

n

=

N

,

X

i

=

]

1

/i

; 0[ for

i

n

,

A

=]

− ∞

; 0[. To show this, it’s enough to prove

d

Z

i

n

Y

i

u ↑

A

=

for

Y

i

=

]

1

/i

; 0[ but this is obvious since

d

Z

i

n

Y

i

=

.

On the other hand,

d

A

i

n

X

i

u ↑

A

6

=

for the same

X

and

A

.

The above theorems are summarized in the diagram at figure

1

:

ID

Strd

A

[

n

]

w

Strd

id

A

[

n

]

= id

Strd

A

[

n

]

ID

Strd

A

[

n

]

w

Strd

id

A

[

n

]

=

id

Strd

A

[

n

]

Figure 1.

Relationships of identity staroids for principal filters.