 21.12. SUBATOMIC PRODUCT OF FUNCOIDS

351

Obvious

1777

.

It is really a filter base.

Obvious

1778

.

Q

RLD

a

w

Q

RLD

a

.

Proposition

1779

.

Q

RLD

a

=

Q

RLD

a

if

n

is finite.

Proof.

Take

m

=

n

to show that

Q

RLD

a

v

Q

RLD

a

.

Proposition

1780

.

Q

RLD

a

=

RLD

(

λi

n

:Base(

a

i

))

if

a

i

is the non-proper filter

for some

i

n

.

Proof.

Take

A

i

=

and

m

=

{

i

}

. Then

Q

i

n

(

A

i

if

i

m

Base(

a

i

) if

i

n

\

m

!

=

.

Example

1781

.

There exists an indexed family

a

of principal filters such that

Q

RLD

a

is non-principal.

Proof.

Let

n

be infinite and Base(

a

i

) is a set of at least two elements. Let

each

a

i

be a trivial ultrafilter.

Every

Q

i

n

(

A

i

if

i

m

Base(

a

i

) if

i

n

\

m

!

has at least 2

n

elements.

There are elements up

Q

RLD

a

with cardinality 1. They can’t be elements of

up

Q

RLD

a

because of cardinality issues.

Corollary

1782

.

There exists an indexed family

a

of principal filters such

that

Q

RLD

a

6

=

Q

RLD

a

.

Proof.

Because

Q

RLD

a

is principal.

Proposition

1783

.

Pr

RLD

k

Q

RLD

x

=

x

k

for every indexed family

x

of proper

filters.

Proof.

Pr

RLD

k

Q

RLD

x

=

h

Pr

k

i

GR

Q

RLD

x

=

x

k

.

Theorem

1784

.

Pr

RLD

i

f

v A

i

for all

i

n

iff

f

v

Q

RLD

A

(for every reloid

f

of arity

n

and

n

-indexed family

A

of filters on sets).

Proof.

f

v

Q

RLD

A ⇒

Pr

RLD

i

f

v

Pr

RLD

i

Q

RLD

A v A

i

.

Let now Pr

RLD

i

f

v A

i

.

f

v

Q

(

Pr

RLD

i

f

if

i

m

Base(form

f

)

i

if

i /

m

!

for finite

m

n

, as it can be easily be

proved by induction.

It follows

f

v

Q

RLD

A

.

21.12. Subatomic product of funcoids

Definition

1785

.

Let

f

be an indexed family of funcoids. Then

Q

(

A

)

f

(

sub-

atomic product

) is a funcoid

Q

i

dom

f

Src

f

i

Q

i

dom

f

Dst

f

i

such that for every

a

atoms

RLD

(

λi

dom

f

:Src

f

i

)

,

b

atoms

RLD

(

λi

dom

f

:Dst

f

i

)

a

(

A

)

Y

f

b

⇔ ∀

i

dom

f

:

RLD

Pr

i

a

[

f

i

]

RLD

Pr

i

b.

Proposition

1786

.

The funcoid

Q

(

A

)

f

exists.