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21.19. IDENTITY STAROIDS AND MULTIFUNCOIDS

372

Proposition

1878

.

Y

GR id

Strd

A

[

n

]

⇔ ∀

A

up

A

:

Y

GR

Strd

id

A

[

n

]

for

every filter

A

on a powerset and

Y

Z

n

.

Proof.

Take

Y

=

Z

n

X

.

A

up

A

:

Y

GR

Strd

id

A

[

n

]

⇔ ∀

A

up

A

:

Z

n

X

GR

Strd

id

A

[

n

]

A

up

A

:

Y

X

6

id

A

[

n

]

⇔↑

Z

n

X

GR id

Strd

A

[

n

]

Y

GR id

Strd

A

[

n

]

.

Proposition

1879

.

Z

n

X

GR id

Strd

a

[

n

]

⇔ ∀

A

a

t

A

i

n

:

t

X

i

.

Proof.

Z

n

X

GR id

Strd

a

[

n

]

⇔ ∃

A

a

t

A

:

n

×{

t

} ∈

Y

X

⇔ ∀

A

a

t

A

i

n

:

t

X

i

.

21.19.5. Relationships between big and small identity staroids.

Definition

1880

.

a

n

Strd

=

Q

Strd

i

n

a

for every element

a

of a poset and an index

set

n

.

Lemma

1881

.

L ∈

GR ID

Strd

a

[

n

]

iff

S

i

n

up

L

i

∪{

a

}

has finite intersection property

(for primary filtrators over meet semilattices with greatest element).

Proof.

The lattice

A

is complete by corollary

515

.

L ∈

GR ID

Strd

a

[

n

]

d

i

n

up

L u

a

6

=

F

⇔ ∀

X

d

i

n

up

L u

a

:

X

6

=

what is equivalent of

S

i

n

L

i

∪ {

a

}

having finite intersection property.

Proposition

1882

.

id

Strd

a

[

n

]

v

ID

Strd

a

[

n

]

v

a

n

Strd

for every filter

a

(on any dis-

tributive lattice with least element) and an index set

n

.

Proof.

GR

id

Strd

a

[

n

]

GR ID

Strd

a

[

n

]

.

L ∈

GR

id

Strd

a

[

n

]

up

L ⊆

GR id

Strd

a

[

n

]

⇔ ∀

L

up

L

:

L

GR id

Strd

a

[

n

]

(theorem

534

)

⇔ ∀

L

up

L∀

A

up

a

:

Z

l

i

n

L

i

6

A

L

up

L∀

A

up

a

:

Z

l

i

n

L

i

u

A

6

=

⊥ ⇒

[

i

n

up

L

i

∪ {

a

}

has finite intersection property

⇔ L ∈

GR ID

Strd

a

[

n

]

.

GR ID

Strd

a

[

n

]

GR

a

n

Strd

.

L ∈

GR ID

Strd

a

[

n

]

MEET

L

i

i

n

 

∪ {

a

}

⇒ ∀

i

a

:

L

i

6

a

⇔ L ∈

GR

a

a

Strd

.

Proposition

1883

.

id

Strd

a

[

a

]

@

ID

Strd

a

[

a

]

=

a

a

Strd

for every nontrivial ultrafilter

a

on a set.

Proof.

GR

id

Strd

a

[

a

]

6

= GR ID

Strd

a

[

a

]

. Let

L

i

=

Base(

a

)

i

. Then trivially

L ∈

GR ID

Strd

a

[

a

]

. But

to disprove

L ∈

GR

id

Strd

a

[

a

]

it’s enough to show

L /

GR id

Strd

a

[

a

]

for some

L

up

L

. Really, take

L

i

=

L

i

=

Base(

a

)

i

. Then

L

GR id

Strd

a

[

a

]

⇔ ∀

A

a

t

A

i

a

:

t

i

what is clearly false (we can always take

i

a

such

that

t /

i

for any point

t

).