background image

Proposition 61.

id

a

[

n

]

Strd

ID

a

[

n

]

Strd

for every filter

a

and an index set

n

.

Proof.

id

a

[

n

]

Strd

=

id

a

[

n

]

Strd

ID

a

[

n

]

Strd

.

Proposition 62.

id

a

[

a

]

Strd

⊏ 

ID

a

[

a

]

Strd

for every nontrivial ultrafilter

a

.

Proof.

Suppose id

a

[

a

]

Strd

=

ID

a

[

a

]

Strd

. Then ID

a

[

a

]

Strd

=

⇈ 

ID

a

[

a

]

Strd

=

id

a

[

a

]

Strd

what contradicts to the

above.

Obvious 63.

L ∈

GR ID

a

[

n

]

Strd

a

d

i

n

L

i

0

F

if

a

is an element of a complete lattice.

Obvious 64.

L ∈

GR ID

a

[

n

]

Strd

⇔ ∀

i

n

:

L

i

a

⇔ ∀

i

n

:

L

i

a

if

a

is an ultrafilter on

A

.

6.6 Identity staroids on principal filters

For principal filter

A

(where

A

is a set) the above definitions coincide with

n

-ary identity relation,

as formulated in the below propositions:

Proposition 65.

Strd

id

A

[

n

]

=

id

A

[

n

]

Strd

.

Proof.

L

GR

Strd

id

A

[

n

]

Q

L

id

A

[

n

]

⇔ ∃

t

A

i

n

:

t

L

i

T

i

n

L

i

A

∅ ⇔

L

GR id

A

[

n

]

Strd

.

Thus

Strd

id

A

[

n

]

=

id

A

[

n

]

Strd

.

Corollary 66.

id

A

[

n

]

Strd

is a principal staroid.

Problem 67.

Is ID

A

[

n

]

Strd

principal for every principal filter

A

on a set and index set

n

?

Proposition 68.

Strd

id

A

[

n

]

ID

A

[

n

]

Strd

for every set

A

.

Proof.

L

GR

Strd

id

A

[

n

]

L

GR id

A

[

n

]

Strd

⇔ ↑

A

d

i

n

A

L

i

⇐ ↑

A

d

i

n

Z

L

i

L

GR ID

A

[

n

]

Strd

.

Proposition 69.

Strd

id

A

[

n

]

⊏ 

ID

A

[

n

]

Strd

for some set

A

and index set

n

.

Proof.

L

GR

Strd

id

A

[

n

]

d

i

n

Z

L

i

A

what is not implied by

d

i

n

A

L

i

A

that is

L

GR ID

A

[

n

]

Strd

. (For a counter example take

n

=

N

,

L

i

= (0; 1/

i

)

,

A

=

R

.)

Proposition 70.

Strd

id

A

[

n

]

=

id

A

[

n

]

Strd

.

Proof.

Strd

id

A

[

n

]

=

id

A

[

n

]

Strd

is obvious from the above.

Proposition 71.

Strd

id

A

[

n

]

ID

A

[

n

]

Strd

.

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

id

A

[

n

]

Strd

⇔ ∀

Y

up

X

:

d

i

n

Z

Y

i

⊓ ↑

A

0

d

i

n

A

X

i

⊓ ↑

A

0

⇔ X ∈

GR ID

A

[

n

]

Strd

.

Proposition 72.

Strd

id

A

[

n

]

ID

A

[

n

]

Strd

for some set

A

.

Proof.

We need to prove

Strd

id

A

[

n

]

ID

A

[

n

]

Strd

that is it’s enough to prove (see the above proof)

that

Y

up

X

:

d

i

n

Z

Y

i

⊓ ↑

A

0

:

d

i

n

A

X

i

⊓ ↑

A

0

. A counter-example follows:

Y

up

X

:

d

i

n

Z

Y

i

⊓ ↑

A

0

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

i

n

Z

Y

i

⊓ ↑

A

0

for

Y

i

=

(

1/

i

; 0)

but this is obvious since

d

i

n

Z

Y

i

= 0

.

On the other hand,

d

i

n

A

X

i

⊓ ↑

A

0

for the same

X

and

A

.

The above theorems are summarized in the following diagram:

Identity staroids and multifuncoids

9