background image

Proof.

If

S

=

;

then

d

Q

RLD

a

j

a

2

S

 

=

d

;

= 1

RLD

(

A

)

and

Q

i

2

dom

A

RLD

d

"

F

(

A

i

)

Pr

i

S

=

Q

i

2

dom

A

RLD

d

"

F

(

A

i

)

;

=

Q

i

2

dom

A

RLD

d

;

=

Q

i

2

dom

A

RLD

1

F

(

A

i

)

= 1

RLD

(

A

)

, thus

d

Q

RLD

a

j

a

2

S

 

=

Q

i

2

dom

A

RLD

d

"

F

(

A

i

)

Pr

i

S:

Let

S

=

/

;

.

d

"

F

(

A

i

)

Pr

i

S

v

d

"

F

(

A

i

)

f

a

i

g

=

a

i

for every

a

2

S

because

a

i

2

Pr

i

S

. Thus

Q

i

2

dom

A

RLD

d

"

F

(

A

i

)

Pr

i

S

v

Q

RLD

a

;

l

(

Y

RLD

a

j

a

2

S

)

w

Y

i

2

dom

A

RLD

l

"

F

(

A

i

)

Pr

i

S:

Now suppose

F

2

GR

Q

i

2

dom

A

RLD

d

"

F

(

A

i

)

Pr

i

S

. Then there exist

X

2

Q

i

2

dom

A

d

"

F

(

A

i

)

Pr

i

S

such that

F

Q

X

. It is enough to prove that there exist

a

2

S

such that

F

2

GR

Q

RLD

a

. For

this it is enough

Q

X

2

GR

Q

RLD

a

.

Really,

X

i

2

d

"

F

(

A

i

)

Pr

i

S

thus

X

i

2

a

i

for every

a

2

S

because Pr

i

S

 f

a

i

g

.

Thus

Q

X

2

GR

Q

RLD

a

.

Denition 17.175.

I call a multireloid

principal

i its graph is a principal lter.

[TODO: Prove

that principal multireloids are the same as multireloid corresponding to a relation.]

Denition 17.176.

I call a multireloid

convex

i it is a join of reloidal products.

Theorem 17.177.

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

)) =

d

"

RLD

(

form

a

)

StarComp

(

A

;

F

)

j

A

2

GR

a;

F

2

Q

i

2

n

GR

f

i

 

t

d

"

RLD

(

form

b

)

StarComp

(

B

;

F

)

j

B

2

GR

b; F

2

Q

i

2

n

GR

f

i

 

=

d

"

RLD

(

form

a

)

StarComp

(

A

;

F

)

t "

RLD

(

form

b

)

StarComp

(

B

;

F

)

j

A

2

GR

a; B

2

GR

b; F

2

Q

i

2

n

GR

f

i

 

=

d

"

RLD

(

form

a

)

(

StarComp

(

A

;

F

)

[

StarComp

(

B

;

F

))

j

A

2

GR

a; B

2

GR

b;

F

2

Q

i

2

n

GR

f

i

 

=

d

"

RLD

(

form

a

)

(

StarComp

(

A

[

B

;

F

))

j

A

2

GR

a; B

2

GR

b; F

2

Q

i

2

n

GR

f

i

 

=

d

"

RLD

(

form

a

)

StarComp

(

C

;

F

)

j

C

2

GR

(

a

t

b

)

; F

2

Q

i

2

n

GR

f

i

 

=

GR StarComp

(

a

t

b

;

f

)

.

Conjecture 17.178.

f

v

Q

RLD

a

, 8

i

2

arity

f

:

Pr

i

RLD

f

v

a

i

for every multireloid

f

and

a

i

2

F

((

form

f

)

i

)

for every

i

2

arity

f

.

17.12.1 Starred reloidal product

Tychono product of topological spaces inspired me the following denition, which seems possibly
useful just like Tychono product:

Denition 17.179.

Let

a

be an

n

-indexed (

n

is an arbitrary index set) family of lters on sets.

Q

RLD

a

(

starred reloidal product

) is the reloid of the form

Q

i

2

n

Base

(

a

i

)

induced by the lter base

(

Y

i

2

n

A

i

if

i

2

m

Base

(

a

i

)

if

i

2

n

n

m

j

m

is a nite subset of

n; A

2

Y

(

a

j

m

)

)

:

Obvious 17.180.

It is really a lter base.

Obvious 17.181.

Q

RLD

a

w

Q

RLD

a

.

Proposition 17.182.

Q

RLD

a

=

Q

RLD

a

if

n

is nite.

Proof.

Take

m

=

n

to show that

Q

RLD

a

v

Q

RLD

a

.

17.12 Multireloids

229