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21.6. JOIN OF MULTIFUNCOIDS

328

Really,

L

i

6

α

i

L

|

(dom

L

)

\{

i

}

L

i

6 h

f

i

i

L

|

(dom

L

)

\{

i

}

t h

g

i

i

L

|

(dom

L

)

\{

i

}

L

i

6 h

f

i

i

L

|

(dom

L

)

\{

i

}

L

i

6 h

g

i

i

L

|

(dom

L

)

\{

i

}

L

j

6 h

f

j

i

L

|

(dom

L

)

\{

j

}

L

j

6 h

g

j

i

L

|

(dom

L

)

\{

j

}

L

j

6 h

f

j

i

L

|

(dom

L

)

\{

j

}

t h

g

j

i

L

|

(dom

L

)

\{

j

}

L

j

6

α

j

L

|

(dom

L

)

\{

j

}

.

Theorem

1682

.

d

pFCD

(

A

)

F

=

d

F

for every set

F

of multifuncoids for the

same indexed family of join infinite distributive complete lattices filtrators.

Proof.

α

i

x

def

=

d

f

F

h

f

i

i

x

. It is enough to prove that

α

is a multifuncoid.

We need to prove:

L

i

6

α

i

L

|

(dom

L

)

\{

i

}

L

j

6

α

j

L

|

(dom

L

)

\{

j

}

.

Really,

L

i

6

α

i

L

|

(dom

L

)

\{

i

}

L

i

6

l

f

F

h

f

i

i

L

|

(dom

L

)

\{

i

}

f

F

:

L

i

6 h

f

i

i

L

|

(dom

L

)

\{

i

}

f

F

:

L

j

6 h

f

j

i

L

|

(dom

L

)

\{

j

}

L

j

6

l

f

F

h

f

j

i

L

|

(dom

L

)

\{

j

}

L

j

6

α

j

L

|

(dom

L

)

\{

j

}

.

Theorem

1683

.

If

f

,

g

are multifuncoids for a primary filtrator (

A

i

,

Z

i

) where

Z

i

are separable starrish posets, then

f

t

pFCD

(

A

)

g

pFCD

(

A

).

Proof.

Let

A

f

t

pFCD

(

A

)

g

and

B

w

A

. Then for every

k

dom

A

A

k

6

f

t

pFCD

(

A

)

g

A

|

(dom

A

)

\{

k

}

;

A

k

6 h

f

t

g

i

A

|

(dom

A

)

\{

k

}

;

A

k

6

h

f

i

(

A

|

(dom

A

)

\{

k

}

)

t h

g

i

(

A

|

(dom

A

)

\{

k

}

).

Thus

A

k

6 h

f

i

(

A

|

(dom

A

)

\{

k

}

)

A

k

6 h

g

i

(

A

|

(dom

A

)

\{

k

}

);

A

[

f

]

A

[

g

]

;

B

[

f

]

B

[

g

]

;

B

k

6 h

f

i

(

B

|

(dom

A

)

\{

k

}

)

B

k

6 h

g

i

(

B

|

(dom

A

)

\{

k

}

);

B

k

6 h

f

i

(

B

|

(dom

A

)

\{

k

}

)

t h

g

i

(

B

|

(dom

A

)

\{

k

}

);

B

k

6 h

f

t

g

i

B

|

(dom

A

)

\{

k

}

=

f

t

pFCD

(

A

)

g

B

|

(dom

A

)

\{

k

}

.

Thus

B

f

t

pFCD

(

A

)

g

.

Theorem

1684

.

If

F

is a set of multifuncoids for the same indexed family of

join infinite distributive complete lattices filtrators, then

d

pFCD

(

A

)

F

pFCD

(

A

).

Proof.

Let

A

h

d

pFCD

(

A

)

F

i

and

B

w

A

. Then for every

k

dom

A

A

k

6

D

d

pFCD

(

A

)

F

E

A

|

(dom

A

)

\{

k

}

=

h

d

F

i

A

|

(dom

A

)

\{

k

}

=

d

f

F

h

f

i

(

A

|

(dom

A

)

\{

k

}

).

Thus

f

F

:

A

k

6 h

f

i

(

A

|

(dom

A

)

\{

k

}

);

f

F

:

A

[

f

]

;

B

[

f

]

for some

f

F

;

f

F

:

B

k

6 h

f

i

(

B

|

(dom

A

)

\{

k

}

);

B

k

6

d

f

F

h

f

i

(

B

|

(dom

A

)

\{

k

}

) =

D

d

pFCD

(

A

)

F

E

B

|

(dom

A

)

\{

k

}

. Thus

B

h

d

pFCD

(

A

)

F

i

.