background image

17.13. SUBATOMIC PRODUCT OF FUNCOIDS

267

Conjecture

1375

.

f

v

Q

RLD

a

⇔ ∀

i

arity

f

: Pr

RLD

i

f

v

a

i

for every

multireloid

f

and

a

i

F

((form

f

)

i

) for every

i

arity

f

.

17.12.1. Starred reloidal product.

Tychonoff product of topological spaces

inspired me the following definition, which seems possibly useful just like Tychonoff

product:

Definition

1376

.

Let

a

be an

n

-indexed (

n

is an arbitrary index set) fam-

ily of filters on sets.

Q

RLD

a

(

starred reloidal product

) is the reloid of the form

Q

i

n

Base(

a

i

) induced by the filter base

Q

i

n

 (

A

i

if

i

m

Base(

a

i

) if

i

n

\

m

!

m

is a finite subset of

n, A

Q

(

a

|

m

)

.

Obvious

1377

.

It is really a filter base.

Obvious

1378

.

Q

RLD

a

w

Q

RLD

a

.

Proposition

1379

.

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

1380

.

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

1381

.

There exists an indexed family

a

of principal filters such that

Q

RLD

a

is non-principal.

Proof.

Let

n

=

N

. Let Base(

a

i

) =

R

and each

a

i

be a principal filter corre-

sponding to a two-element set.

Every

Q

i

n

 (

A

i

if

i

m

Base(

a

i

) if

i

n

\

m

!

has at least

c

n

>

c

elements.

There are elements

Q

RLD

a

with cardinality 2

n

=

n

. They can’t be elements of

Q

RLD

a

because

n

=

ω <

c

.

Corollary

1382

.

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

1383

.

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

.

17.13. Subatomic product of funcoids

Definition

1384

.

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.