X

a, Y

b

i

dom

f

:

Pr

i

X

[

f

i

]

Pr

i

Y

.

Then

X

∈ h

Pr

i

i

a, Y

∈ h

Pr

i

i

b

:

X

[

f

i

]

Y

.

Thus by the lemma

X

d

h↑

Src

f

i

ih

Pr

i

i

a, Y

d

h↑

Dst

f

i

ih

Pr

i

i

b

:

X

[

f

i

]

Y

.

X

Pr

i

a, Y

Pr

i

b

:

X

[

f

i

]

Y

.

Thus Pr

i

a

[

f

i

]

Pr

i

b

. So

i

dom

f

:

Pr

i

a

[

f

i

]

Pr

i

b

and thus

a

f

×

(

A

)

g

b

.

Remark 171.

It seems that the proof of the above theorem can be simplified using cross-compo-

sition product.

Theorem 172.

Q

i

n

(

A

)

(

g

i

f

i

) =

Q

(

A

)

g

Q

(

A

)

f

for indexed (by an index set

n

) families

f

and

g

of funcoids such that

i

n

:

Dst

f

i

=

Src

g

i

.

Proof.

Let

a

,

b

are ultrafilters on

Q

i

n

Src

f

i

and

Q

i

n

Dst

g

i

correspondingly,

a

"

Y

i

n

(

A

)

(

g

i

f

i

)

#

b

⇔ ∀

i

dom

f

:

Pr

i

a

[

g

i

f

i

]

Pr

i

b

⇔ ∀

i

dom

f

C

atoms

F

Q

i

n

Dst

f i

:

(

Pr

i

a

[

f

i

]

C

C

[

g

i

]

Pr

i

b

)

⇔ ∀

i

dom

f

c

atoms

RLD

(

λi

n

:

Dst

f

)

: (

Pr

i

a

[

f

i

]

Pr

i

c

Pr

i

c

[

g

i

]

Pr

i

b

)

c

atoms

RLD

(

λi

n

:

Dst

f

)

i

dom

f

: (

Pr

i

a

[

f

i

]

Pr

i

c

Pr

i

c

[

g

i

]

Pr

i

b

)

⇔ ∃

c

atoms

RLD

(

λi

n

:

Dst

f

)

:

a

"

Y

(

A

)

f

#

c

c

"

Y

(

A

)

g

#

b

a

"

Y

(

A

)

g

Y

(

A

)

f

#

b.

Let

i

dom

f

c

atoms

RLD

(

λi

n

:

Dst

f

)

: (

Pr

i

a

[

f

i

]

Pr

i

c

Pr

i

c

[

g

i

]

Pr

i

b

)

.

Then there exists

c

atoms

RLD

(

λi

n

:

Dst

f

)

such that

i

dom

f

: (

Pr

i

a

[

f

i

]

Pr

i

c

i

Pr

i

c

i

[

g

i

]

Pr

i

b

)

.

Then take

c

′′

=

Q

RLD

c

. Then

i

dom

f

: (

Pr

i

a

[

f

i

]

Pr

i

c

i

′′

Pr

i

c

i

′′

[

g

i

]

Pr

i

b

)

. Thus

c

atoms

RLD

(

λi

n

:

Dst

f

)

i

dom

f

: (

Pr

i

a

[

f

i

]

Pr

i

c

Pr

i

c

[

g

i

]

Pr

i

b

)

.

We have

a

h

Q

i

n

(

A

)

(

g

i

f

i

)

i

b

a

h

Q

(

A

)

g

Q

(

A

)

f

i

b

.

Proposition 173.

Q

RLD

a

h

Q

(

A

)

f

i

Q

RLD

b

⇔ ∀

i

dom

f

:

a

i

[

f

i

]

b

i

for an indexed family

f

of

funcoids and indexed families

a

abd

b

of filters where

a

i

F

(

Src

f

)

,

b

i

F

(

Dst

f

)

for every

i

dom

f

.

Proof.

Q

RLD

a

h

Q

(

A

)

f

i

Q

RLD

b

⇔ ∃

x

atoms

Q

RLD

a, y

atoms

Q

RLD

b

:

x

h

Q

(

A

)

f

i

y

x

atoms

Q

RLD

a, y

atoms

Q

RLD

b

i

dom

f

:

Pr

i

x

[

f

i

]

Pr

i

y

⇔ ∃

x

atoms

Q

RLD

a,

y

atoms

Q

RLD

b

i

dom

f

:

a

i

[

f

i

]

b

i

⇔ ∀

i

dom

f

:

a

i

[

f

i

]

b

i

.

15 On products and projections

Conjecture 174.

For discrete funcoids

Q

(

C

)

and

Q

(

A

)

coincide with the conventional product

of binary relations.

15.1 Staroidal product

Let

f

is a staroid components of whose form are boolean lattices.

Definition 175.

Staroidal projection

of a staroid

Pr

k

Strd

f

=

h

f

i

k

λi

(

arity

f

)

\ {

k

}

: 1

(

form

f

)

i

.

28

Section 15