Proof.

Special case when

S

is empty is obvious. Let

S

.

d

F

(

A

i

)

Pr

i

S

d

F

(

A

i

)

{

a

i

}

=

a

i

for every

a

S

because

a

i

Pr

i

S

. Thus

Q

i

dom

A

RLD

d

F

(

A

i

)

Pr

i

S

Q

RLD

a

;

l

(

Y

RLD

a

|

a

S

)

Y

i

dom

A

RLD

l

F

(

A

i

)

Pr

i

S.

Now suppose

F

Q

i

dom

A

RLD

d

F

(

A

i

)

Pr

i

S

. Then there exist

X

λi

dom

A

:

d

F

(

A

i

)

Pr

i

S

such that

F

Q

X

. It is enough to prove that there exist

a

S

such that

F

Q

RLD

a

. For this

it is enough

Q

X

Q

RLD

a

.

Really,

X

i

d

F

(

A

i

)

Pr

i

S

thus

X

i

a

i

for every

a

S

because Pr

i

S

⊇ {

a

i

}

.

Thus

Q

X

Q

RLD

a

.

Definition 166.

I call a multireloid

convex

iff it is a join of reloidal products.

Conjecture 167.

f

Q

RLD

a

⇔ ∀

i

arity

f

:

Pr

i

RLD

f

a

i

for every multireloid

f

and

a

i

F

((

form

f

)

i

)

for every

i

arity

f

.

14 Subatomic product of funcoids

Lemma 168.

d

h↑

A

ih

Pr

i

i

a

=

h

Pr

i

i

a

for every multireloid

a

and

i

arity

a

.

Proof.

d

A

h

Pr

i

i

a

⊇ h

Pr

i

i

a

is obvious.

h

Pr

i

i

a

is a filter base. Really, let

P , Q

∈ h

Pr

i

i

a

. Then

P

=

dom

X

0

,

Q

=

dom

X

1

where

X

0

,

X

1

a

. Then

P

Q

=

dom

X

0

dom

X

1

dom

(

X

0

X

1

)

∈ h

Pr

i

i

a

.

Let

K

d

h↑

A

ih

Pr

i

i

a

. Then by properties of generalized filter bases there exists

X

a

such

that

K

⊇ h↑

A

ih

Pr

i

i

X

that is

K

Pr

i

X

and consequently

K

∈ h

Pr

i

i

a

.

Definition 169.

Let

f

is an indexed family of funcoids. Then

Q

(

A

)

f

(

subatomic product

) is

a funcoid

Q

i

dom

f

Src

f

i

Q

i

dom

f

Dst

f

i

such that for every

a

atoms

1

RLD

(

λi

dom

f

:

Src

f

i

)

,

b

atoms

1

RLD

(

λi

dom

f

:

Dst

f

i

)

a

"

Y

(

A

)

f

#

b

⇔ ∀

i

dom

f

:

Pr

i

a

[

f

i

]

Pr

i

b.

Proposition 170.

The funcoid

Q

(

A

)

f

exists.

Proof.

To prove that

Q

(

A

)

f

exists we need to prove (for every

a

atoms

1

RLD

(

λi

dom

f

:

Src

f

i

)

,

b

atoms

1

RLD

(

λi

dom

f

:

Dst

f

i

)

)

X

a, Y

b

x

atoms

RLD

(

λi

dom

f

:

Src

f

i

)

X , y

atoms

RLD

(

λi

dom

f

:

Dst

f

i

)

Y

:

x

"

Y

(

A

)

f

#

y

a

"

Y

(

A

)

f

#

b.

Let

X

a, Y

b

x

atoms

RLD

(

λi

dom

f

:

Src

f

i

)

X , y

atoms

RLD

(

λi

dom

f

:

Dst

f

i

)

Y

:

x

h

Q

(

A

)

f

i

y

.

Then

X

a, Y

b

x

atoms

RLD

(

λi

dom

f

:

Src

f

i

)

X , y

atoms

RLD

(

λi

dom

f

:

Dst

f

i

)

Y

i

dom

f

:

Pr

i

x

[

f

i

]

Pr

i

y.

Then because Pr

i

x

atoms

Src

f

i

Pr

i

X

and likewise for

y

:

Then

X

a, Y

b

i

dom

f

x

atoms

Src

f

i

Pr

i

X , y

atoms

Dst

f

i

Pr

i

Y

:

x

[

f

i

]

y

.

Thus

X

a, Y

b

i

dom

f

:

Src

f

i

Pr

i

X

[

f

i

]

Dst

f

i

Pr

i

Y

;

Subatomic product of funcoids

27