background image

Proof.

L 2

ID

a

[

n

]

Strd

,

up

ID

a

[

n

]

Strd

,

up

ID

a

[

n

]

Strd

, 8

L

2

up

L

:

L

2

ID

a

[

n

]

Strd

, 8

L

2

up

L

:

d

i

2

n

L

i

u

a

=

/ 0

F

,

S

i

2

n

L

i

[

a

has nite intersection property

,

(lemma)

,L 2

GR ID

a

[

n

]

Strd

.

Proposition 18.61.

id

a

[

n

]

Strd

v

ID

a

[

n

]

Strd

for every lter

a

and an index set

n

.

Proof.

id

a

[

n

]

Strd

=

id

a

[

n

]

Strd

v

ID

a

[

n

]

Strd

.

Proposition 18.62.

id

a

[

a

]

Strd

ID

a

[

a

]

Strd

for every nontrivial ultralter

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 18.63.

L 2

GR ID

a

[

n

]

Strd

,

a

u

d

i

2

n

L

i

=

/ 0

F

if

a

is an element of a complete lattice.

Obvious 18.64.

L 2

GR ID

a

[

n

]

Strd

, 8

i

2

n

:

L

i

w

a

, 8

i

2

n

:

L

i

/

a

if

a

is an ultralter on

A

.

18.4.6 Identity staroids on principal lters

For principal lter

"

A

(where

A

is a set) the above denitions coincide with

n

-ary identity relation,

as formulated in the following propositions:

Proposition 18.65.

"

Strd

id

A

[

n

]

=

id

"

A

[

n

]

Strd

.

Proof.

L

2

GR

"

Strd

id

A

[

n

]

,

Q

L

/

id

A

[

n

]

, 9

t

2

A

8

i

2

n

:

t

2

L

i

,

T

i

2

n

L

i

\

A

=

/

; ,

L

2

GR id

"

A

[

n

]

Strd

.

Thus

"

Strd

id

A

[

n

]

=

id

"

A

[

n

]

Strd

.

Corollary 18.66.

id

"

A

[

n

]

Strd

is a principal staroid.

Question 18.67.

Is ID

A

[

n

]

Strd

principal for every principal lter

A

on a set and index set

n

?

Proposition 18.68.

"

Strd

id

A

[

n

]

v

ID

"

A

[

n

]

Strd

for every set

A

.

Proof.

L

2

GR

"

Strd

id

A

[

n

]

,

L

2

GR id

"

A

[

n

]

Strd

, "

A

/

d

i

2

n

A

L

i

( "

A

/

d

i

2

n

Z

L

i

,

L

2

GR ID

"

A

[

n

]

Strd

.

Proposition 18.69.

"

Strd

id

A

[

n

]

ID

"

A

[

n

]

Strd

for some set

A

and index set

n

.

Proof.

L

2

GR

"

Strd

id

A

[

n

]

,

d

i

2

n

Z

L

i

/

"

A

what is not implied by

d

i

2

n

A

L

i

/

"

A

that is

L

2

GR ID

"

A

[

n

]

Strd

. (For a counter example take

n

=

N

,

L

i

= (0; 1/

i

)

,

A

=

R

.)

Proposition 18.70.

"

Strd

id

A

[

n

]

=

id

"

A

[

n

]

Strd

.

Proof.

"

Strd

id

A

[

n

]

=

id

"

A

[

n

]

Strd

is obvious from the above.

Proposition 18.71.

"

Strd

id

A

[

n

]

v

ID

"

A

[

n

]

Strd

.

Proof.

X 2

GR

"

Strd

id

A

[

n

]

,

up

GR

"

Strd

id

A

[

n

]

, 8

Y

2

up

X

:

Y

2

GR

"

Strd

id

A

[

n

]

, 8

Y

2

up

X

:

Y

2

id

"

A

[

n

]

Strd

, 8

Y

2

up

X

:

d

i

2

n

Z

Y

i

u "

A

=

/ 0

)

d

i

2

n

A

X

i

u "

A

=

/ 0

, X 2

GR ID

"

A

[

n

]

Strd

.

Proposition 18.72.

"

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

8

Y

2

up

X

:

d

i

2

n

Z

Y

i

u "

A

=

/ 0

:

d

i

2

n

A

X

i

u "

A

=

/ 0

. A counter-example follows:

18.4 Identity staroids and multifuncoids

249