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Remark 82.

To describe this, the definition of order poset is used twice. Let

f

and

g

are posets

of the same form

A

h

f

i ⊑ h

g

i ⇔ ∀

i

dom

A

:

h

f

i

i

⊑ h

g

i

i

and

h

f

i

i

⊑ h

g

i

i

⇔ ∀

L

Y

A

|

(

dom

A

)

\{

i

}

:

h

f

i

i

L

⊑ h

g

i

i

L.

Theorem 83.

f

pFCD

(

A

)

g

=

f

g

for every pre-multifuncoids

f

and

g

of the same form

A

of

distributive lattices.

Proof.

α

i

x

=

def

f

i

x

g

i

x

. It is enough to prove that

α

is a multifuncoid.

We need to prove:

L

i

α

i

L

|

(

dom

L

)

\{

i

}

L

j

α

j

L

|

(

dom

L

)

\{

j

}

.

Really,

L

i

α

i

L

|

(

dom

L

)

\{

i

}

L

i

f

i

L

|

(

dom

L

)

\{

i

}

g

i

L

|

(

dom

L

)

\{

i

}

L

i

f

i

L

|

(

dom

L

)

\{

i

}

L

i

g

i

L

|

(

dom

L

)

\{

i

}

L

j

f

j

L

|

(

dom

L

)

\{

j

}

L

j

g

j

L

|

(

dom

L

)

\{

j

}

L

j

f

j

L

|

(

dom

L

)

\{

j

}

g

j

L

|

(

dom

L

)

\{

j

}

L

j

α

j

L

|

(

dom

L

)

\{

j

}

.

Theorem 84.

F

pFCD

(

A

)

F

=

F

F

for every set

F

of pre-multifuncoids of the same form

A

of join

infinite distributive complete lattices.

Proof.

α

i

x

=

def

F

f

F

f

i

x

. It is enough to prove that

α

is a multifuncoid.

We need to prove:

L

i

α

i

L

|

(

dom

L

)

\{

i

}

L

j

α

j

L

|

(

dom

L

)

\{

j

}

.

Really,

L

i

α

i

L

|

(

dom

L

)

\{

i

}

L

i

F

f

F

f

i

L

|

(

dom

L

)

\{

i

}

⇔∃

f

F

:

L

i

f

i

L

|

(

dom

L

)

\{

i

}

⇔∃

f

F

:

L

j

f

j

L

|

(

dom

L

)

\{

j

}

L

j

F

f

F

f

j

L

|

(

dom

L

)

\{

j

}

L

j

α

j

L

|

(

dom

L

)

\{

j

}

.

Proposition 85.

The mapping

f

[

f

]

is an order embedding, for multifuncoids of the form

A

of

separable starrish posets.

Proof.

The mapping

f

[

f

]

is defined because

A

are starrish poset. The mapping is injective

because

A

are separable posets. That

f

[

f

]

is a monotone function is obvious.

Remark 86.

This order embedding is useful to describe properties of posets of pre-staroids.

Theorem 87.

If

f

,

g

are multifuncoids of the same form

A

of distributive lattices, then

f

pFCD

(

A

)

g

FCD

(

A

)

.

Proof.

Let

A

f

pFCD

(

A

)

g

and

B

A

. Then for every

k

dom

A

A

k

f

pFCD

(

A

)

g

A

|

(

dom

A

)

\{

k

}

=(

f

g

)

A

|

(

dom

A

)

\{

k

}

=

f

(

A

|

(

dom

A

)

\{

k

}

)

g

(

A

|

(

dom

A

)

\{

k

}

)

.

Thus

A

k

f

(

A

|

(

dom

A

)

\{

k

}

)

A

k

g

(

A

|

(

dom

A

)

\{

k

}

)

;

A

[

f

]

A

[

g

]

;

B

[

f

]

B

[

g

]

;

B

k

f

(

B

|

(

dom

A

)

\{

k

}

)

B

k

g

(

B

|

(

dom

A

)

\{

k

}

)

;

f

(

B

|

(

dom

A

)

\{

k

}

)

g

(

B

|

(

dom

A

)

\{

k

}

) = (

f

g

)

B

|

(

dom

A

)

\{

k

}

=

f

pFCD

(

A

)

g

B

|

(

dom

A

)

\{

k

}

B

k

. Thus

B

f

pFCD

(

A

)

g

.

Theorem 88.

If

F

is a set multifuncoids of the same form

A

of join inifinite distributive complete

lattices, then

F

pFCD

(

A

)

f

FCD

(

A

)

.

Proof.

Let

A

h

F

pFCD

(

A

)

f

i

and

B

A

. Then for every

k

dom

A

.

A

k

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

f

(

A

|

(

dom

A

)

\{

k

}

)

;

f

F

:

A

[

f

]

;

B

[

f

]

B

[

g

]

;

f

F

:

B

k

f

(

B

|

(

dom

A

)

\{

k

}

)

;

F

f

F

f

(

B

|

(

dom

A

)

\{

k

}

) = (

f

g

)

B

|

(

dom

A

)

\{

k

}

=

F

pFCD

(

A

)

F

B

|

(

dom

A

)

\{

k

}

B

k

.

Thus

B

h

F

pFCD

(

A

)

F

i

.

Conjecture 89.

The formula

f

FCD

(

A

)

g

cFCD

(

A

)

is not true in general for completary

multifuncoids (even for multifuncoids on powersets)

f

and

g

of the same form

A

.

Join of multifuncoids

11