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17.7. JOIN OF MULTIFUNCOIDS

245

Theorem

1288

.

F

pFCD

(

A

)

F

=

F

F

for every set

F

of premultifuncoids for the

same indexed family of join infinite distributive complete lattices filtrators.

Proof.

α

i

x

def

=

F

f

F

f

i

x

. It is enough to prove that

α

is a premultifuncoid.

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

G

f

F

f

i

L

|

(dom

L

)

\{

i

}

f

F

:

L

i

6

f

i

L

|

(dom

L

)

\{

i

}

f

F

:

L

j

6

f

j

L

|

(dom

L

)

\{

j

}

L

j

6

G

f

F

f

j

L

|

(dom

L

)

\{

j

}

L

j

6

α

j

L

|

(dom

L

)

\{

j

}

.

Proposition

1289

.

The mapping

f

7→

[

f

] is an order embedding, for multifun-

coids for indexed families (

A

i

;

Z

i

) of down-aligned starrish filtrators with separable

finitely meet-closed core.

Proof.

The mapping

f

7→

[

f

] is defined because

A

i

are starrish posets (and

(

A

i

;

Z

i

) is with finitely meet-closed core and down-aligned). The mapping is injec-

tive because the filtrators are with separable cores (

n

X

Z

i

X

6h

f

i

A

o

=

n

X

Z

i

X

6h

f

i

B

o

implies

h

f

i

A

=

h

f

i

B

). That

f

7→

[

f

] is a monotone function is obvious.

Remark

1290

.

This order embedding is useful to describe properties of posets

of prestaroids.

Theorem

1291

.

If

f

,

g

are multifuncoids for the filtrator (

F

i

;

P

i

) where

Z

i

are

separable starrish posets, then

f

t

pFCD

(

A

)

g

FCD

(

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

(

f

t

g

)

A

|

(dom

A

)

\{

k

}

;

A

k

6

f

(

A

|

(dom

A

)

\{

k

}

)

t

g

(

A

|

(dom

A

)

\{

k

}

).

Thus

A

k

6

f

(

A

|

(dom

A

)

\{

k

}

)

A

k

6

g

(

A

|

(dom

A

)

\{

k

}

);

A

[

f

]

A

[

g

];

B

[

f

]

B

[

g

];

B

k

6

f

(

B

|

(dom

A

)

\{

k

}

)

B

k

6

g

(

B

|

(dom

A

)

\{

k

}

);

B

k

6

f

(

B

|

(dom

A

)

\{

k

}

)

t

g

(

B

|

(dom

A

)

\{

k

}

);

B

k

6

(

f

t

g

)

B

|

(dom

A

)

\{

k

}

=

(

f

t

pFCD

(

A

)

g

)

B

|

(dom

A

)

\{

k

}

.

Thus

B

f

t

pFCD

(

A

)

g

.

Theorem

1292

.

If

F

is a set of multifuncoids for the same indexed family of

join infinite distributive complete lattices filtrators, then

F

pFCD

(

A

)

F

FCD

(

A

).

Proof.

Let

A

h

F

pFCD

(

A

)

F

i

and

B

w

A

. Then for every

k

dom

A

A

k

6

F

pFCD

(

A

)

F

A

|

(dom

A

)

\{

k

}

=

(

F

F

)

A

|

(dom

A

)

\{

k

}

=

F

f

F

f

(

A

|

(dom

A

)

\{

k

}

).

Thus

f

F

:

A

k

6

f

(

A

|

(dom

A

)

\{

k

}

);

f

F

:

A

[

f

];

B

[

f

] for

some

f

F

;

f

F

:

B

k

6

f

(

B

|

(dom

A

)

\{

k

}

);

B

k

6

F

f

F

f

(

B

|

(dom

A

)

\{

k

}

) =

F

pFCD

(

A

)

F

B

|

(dom

A

)

\{

k

}

. Thus

B

h

F

pFCD

(

A

)

F

i

.