background image

18.4. IDENTITY STAROIDS AND MULTIFUNCOIDS

292

Corollary

1509

.

Staroidal product of an infinite indexed family of ultrafilters

may be non-atomic.

Proposition

1510

.

id

Strd

a

[

n

]

is determined by the value of

id

Strd

a

[

n

]

. Moreover

id

Strd

a

[

n

]

=

id

Strd

a

[

n

]

.

Proof.

Use general properties of upgrading and downgrading (proposition

1261

).

Lemma

1511

.

L ∈

GR ID

Strd

a

[

n

]

iff

S

i

n

L

i

a

has finite intersection property

(for primary filtrators).

Proof.

L ∈

GR ID

Strd

a

[

n

]

d

i

n

L u

a

6

= 0

F

⇔ ∀

X

d

i

n

L u

a

:

X

6

=

what

is equivalent of

S

i

n

L

i

a

having finite intersection property.

Proposition

1512

.

ID

Strd

a

[

n

]

is determined by the value of

ID

Strd

a

[

n

]

, moreover

ID

Strd

a

[

n

]

=

ID

Strd

a

[

n

]

(for primary filtrators).

Proof.

L ∈

ID

Strd

a

[

n

]

up

L ⊆

ID

Strd

a

[

n

]

up

L ⊆

ID

Strd

a

[

n

]

L

up

L

:

L

ID

Strd

a

[

n

]

⇔ ∀

L

up

L

:

l

i

n

L

i

u

a

6

= 0

F

[

i

n

L

i

a

has finite intersection property

(lemma)

⇔ L ∈

GR ID

Strd

a

[

n

]

.

Proposition

1513

.

id

Strd

a

[

n

]

v

ID

Strd

a

[

n

]

for every filter

a

and an index set

n

.

Proof.

id

Strd

a

[

n

]

=

id

Strd

a

[

n

]

v

ID

Strd

a

[

n

]

.

Proposition

1514

.

id

Strd

a

[

a

]

@

ID

Strd

a

[

a

]

for every nontrivial ultrafilter

a

.

Proof.

Suppose id

Strd

a

[

a

]

=

ID

Strd

a

[

a

]

. Then ID

Strd

a

[

a

]

=

ID

Strd

a

[

a

]

=

id

Strd

a

[

a

]

what

contradicts to the above.

Obvious

1515

.

L ∈

GR ID

Strd

a

[

n

]

a

u

d

i

n

L

i

6

=

F

if

a

is an element of a

complete lattice.

Obvious

1516

.

L ∈

GR ID

Strd

a

[

n

]

⇔ ∀

i

n

:

L

i

w

a

⇔ ∀

i

n

:

L

i

6

a

if

a

is an

ultrafilter on

A

.

18.4.6. 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 following propositions:

Proposition

1517

.

Strd

id

A

[

n

]

= id

Strd

A

[

n

]

.

Proof.

L

GR

Strd

id

A

[

n

]

Y

L

6

id

A

[

n

]

⇔ ∃

t

A

i

n

:

t

L

i

\

i

n

L

i

A

6

=

∅ ⇔

L

GR id

Strd

A

[

n

]

.

Thus

Strd

id

A

[

n

]

= id

Strd

A

[

n

]

.

Corollary

1518

.

id

Strd

A

[

n

]

is a principal staroid.