background image

2.

8

i

2

n

:

f

i

2

C

0

(

i

;

i

)

)

Q

(

S

)

f

2

C

0

 Q

(

S

)

;

Q

(

S

)

;

3.

8

i

2

n

:

f

i

2

C

00

(

i

;

i

)

)

Q

(

S

)

f

2

C

00

 Q

(

S

)

;

Q

(

S

)

.

Proof.

Similar to the previous theorem.

17.17 Counter-examples

Example 17.227.

f

=

/

f

for some staroid

f

whose form is an indexed family of lters on a set.

Proof.

Let

f

=

fA 2

F

(

f

)

j "

f

Cor

g

for some innite set

f

where

is some non-principal

lter on

f

.

A

t

B

2

f

, "

f

Cor

(

A

t

B

)

, "

f

Cor

A

t "

f

Cor

B

, "

f

Cor

A

u

 =

/ 0

F

(

f

)

_

"

f

Cor

B

u

 =

/ 0

F

(

f

)

,

A

2

f

_

B

2

f

.

Obviously

0

F

(

f

)

2

/

f

. So

f

is a free star. But free stars are essentially the same as

1

-staroids.

f

=

@

.

f

=

?

 =

/

f

.

For the below counter-examples we will dene a staroid

#

with arity

#

=

N

and GR

#

2

P

(

N

N

)

(based on a suggestion by Andreas Blass):

A

2

GR

#

,

sup

i

2

N

card

(

A

i

\

i

) =

N

^ 8

i

2

N

:

A

i

=

/

;

:

Proposition 17.228.

#

is a staroid.

Proof.

(

val

#

)

i

L

=

P

N

n f;g

for every

L

2

(

P

N

)

N

nf

i

g

if

sup

i

2

N

nf

i

g

card

(

A

j

\

j

) =

N

^ 8

j

2

N

n f

i

g

:

L

j

=

/

;

:

Otherwise

(

val

#

)

i

L

=

;

. Thus

(

val

#

)

i

L

is a free star. So

#

is a staroid.

[TODO: Show that it's not

just a prestaroid.]

Proposition 17.229.

#

is a completary staroid.

Proof.

A

0

t

A

1

2

GR

#

,

A

0

[

A

1

2

GR

#

,

sup

i

2

N

card

((

A

0

i

[

A

1

i

)

\

i

) =

N

^ 8

i

2

N

:

A

0

i

[

A

1

i

=

/

; ,

sup

i

2

N

card

((

A

0

i

\

i

)

[

(

A

1

i

\

i

)) =

N

^ 8

i

2

N

:

A

0

i

[

A

1

i

=

/

;

.

If

A

0

i

=

;

then

A

0

i

\

i

=

;

and thus

A

1

i

\

i

w

A

0

i

\

i

. Thus we can select

c

(

i

)

2 f

0

;

1

g

in such a

way that

8

d

2 f

0

;

1

g

:

card

(

A

c

(

i

)

i

\

i

)

w

card

(

A

d

i

\

i

)

and

A

c

(

i

)

i

=

/

;

. (Consider the case

A

0

i; A

1

i

=

/

;

and the similar cases

A

0

i

=

;

and

A

1

i

=

;

.)

So

A

0

t

A

1

2

GR

#

,

sup

i

2

N

card

(

A

c

(

i

)

i

\

i

) =

N

^ 8

i

2

N

:

A

c

(

i

)

i

=

/

; ,

(

i

2

n

:

A

c

(

i

)

i

)

2

GR

#

.

Thus

#

is completary.

Obvious 17.230.

#

is non-zero.

Example 17.231.

For every family

a

=

a

i

2

N

of ultralters

Q

Strd

a

is not an atom nor of the poset

of staroids neither of the poset of completary staroids of the form

i

2

N

:

Base

(

a

i

)

.

Proof.

It's enough to prove

#

w

Q

Strd

a

.

Let

"

N

R

i

=

a

i

if

a

i

is principal and

R

i

=

N

n

i

if

a

i

is non-principal.

We have

8

i

2

N

:

R

i

2

a

i

.

We have

R

2

/

GR

#

because sup

i

2

N

card

(

R

i

\

i

) =

/

N

.

R

2

Q

Strd

a

because

8

X

2

a

i

:

X

\

R

i

=

/

;

.

So

#

w

Q

Strd

a

.

Remark 17.232.

At http://mathoverow.net/questions/60925/special-innitary-relations-and-

ultralters there are a proof for arbitrary innite form, not just for

N

.

17.17 Counter-examples

237