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21.11. MULTIRELOIDS

350

thus

d

a

S

Q

RLD

a

=

Q

RLD

i

dom

Z

d

F

(

Z

i

)

Pr

i

S

.

Let

S

6

=

.

d

F

(

Z

i

)

Pr

i

S

v

d

F

(

Z

i

)

{

a

i

}

=

a

i

for every

a

S

because

a

i

Pr

i

S

. Thus

Q

RLD

i

dom

Z

d

F

(

Z

i

)

Pr

i

S

v

Q

RLD

a

;

l

a

S

RLD

Y

a

w

RLD

Y

i

dom

Z

F

(

Z

i

)

l

Pr

i

S.

Now suppose

F

GR

Q

RLD

i

dom

Z

d

F

(

Z

i

)

Pr

i

S

.

Then there exists

X

up

Q

i

dom

Z

d

F

(

Z

i

)

Pr

i

S

such that

F

Q

X

. It is enough to prove that there

exist

a

S

such that

F

GR

Q

RLD

a

. For this it is enough

Q

X

GR

Q

RLD

a

.

Really,

X

i

up

d

F

(

Z

i

)

Pr

i

S

thus

X

i

up

a

i

for every

A

S

because Pr

i

S

{

a

i

}

.

Thus

Q

X

GR

Q

RLD

a

.

Definition

1770

.

I call a multireloid

principal

iff its graph is a principal filter.

Definition

1771

.

I call a multireloid

convex

iff it is a join of reloidal products.

Theorem

1772

.

StarComp(

a

t

b, f

) = StarComp(

a, f

)

t

StarComp(

b, f

) for

multireloids

a

,

b

and an indexed family

f

of reloids with Src

f

i

= (form

a

)

i

=

(form

b

)

i

.

Proof.

GR(StarComp(

a, f

)

t

StarComp(

b, f

)) =

l

RLD

(form

a

)

StarComp(

A, F

)

A

GR

a, F

Q

i

n

GR

f

i

t

l

RLD

(form

b

)

StarComp(

B, F

)

B

GR

b, F

Q

i

n

GR

f

i

=

l

RLD

(form

a

)

StarComp(

A, F

)

t ↑

RLD

(form

b

)

StarComp(

B, F

)

A

GR

a, B

GR

b, F

Q

i

n

GR

f

i

=

l

RLD

(form

a

)

(StarComp(

A, F

)

StarComp(

B, F

))

A

GR

a, B

GR

b, F

Q

i

n

GR

f

i

=

l

RLD

(form

a

)

StarComp(

A

B, F

)

A

GR

a, B

GR

b, F

Q

i

n

GR

f

i

=

l

RLD

(form

a

)

StarComp(

C, F

)

C

GR(

a

t

b

)

, F

Q

i

n

GR

f

i

=

GR StarComp(

a

t

b, f

)

.

21.11.1. Starred reloidal product.

Tychonoff product of topological spaces

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

product:

Definition

1773

.

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

)

.