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21.17. UPGRADING AND DOWNGRADING MULTIFUNCOIDS

362

Theorem

1828

.

Let

µ

and

ν

be indexed (by some index set

n

) families of

endofuncoids, and

f

i

FCD

(Ob

µ

i

,

Ob

ν

i

) for every

i

n

. Then:

1

.

i

n

:

f

i

C(

µ

i

, ν

i

)

Q

(

A

)

f

C

Q

(

A

)

µ,

Q

(

A

)

ν

;

2

.

i

n

:

f

i

C

0

(

µ

i

, ν

i

)

Q

(

A

)

f

C

0

Q

(

A

)

µ,

Q

(

A

)

ν

;

3

.

i

n

:

f

i

C

00

(

µ

i

, ν

i

)

Q

(

A

)

f

C

00

Q

(

A

)

µ,

Q

(

A

)

ν

.

Proof.

Similar to the previous theorem.

Theorem

1829

.

Let

µ

and

ν

be indexed (by some index set

n

) families of point-

free endofuncoids between posets with least elements, and

f

i

pFCD

(Ob

µ

i

,

Ob

ν

i

)

for every

i

n

. Then:

1

.

i

n

:

f

i

C(

µ

i

, ν

i

)

Q

(

S

)

f

C

Q

(

S

)

µ,

Q

(

S

)

ν

;

2

.

i

n

:

f

i

C

0

(

µ

i

, ν

i

)

Q

(

S

)

f

C

0

Q

(

S

)

µ,

Q

(

S

)

ν

;

3

.

i

n

:

f

i

C

00

(

µ

i

, ν

i

)

Q

(

S

)

f

C

00

Q

(

S

)

µ,

Q

(

S

)

ν

.

Proof.

Similar to the previous theorem.

21.17. Upgrading and downgrading multifuncoids

Lemma

1830

.

(

h

f

i

k

X

X

up

Q

i

n

\{

k

}

Z

i

X

)

is a filter base on

A

k

for every family

(

A

i

,

Z

i

) of primary filtrators where

i

n

for some index set

n

(provided that

f

is a

multifuncoid of the form

Z

and

k

n

and

X ∈

Q

i

n

\{

k

}

A

i

).

Proof.

Let

K

,

L ∈

n

h

f

i

k

X

X

up

X

o

. Then there exist

X, Y

up

X

such that

K

=

h

f

i

k

X

,

L

=

h

f

i

k

Y

. We can take

Z

up

X

such that

Z

v

X, Y

. Then evidently

h

f

i

k

Z

v K

and

h

f

i

k

Z

v L

and

h

f

i

k

Z

n

h

f

i

k

X

X

up

X

o

.

Definition

1831

.

Square

mult is a mult whose base and core are the same.

Definition

1832

.

L ∈

[

f

]

⇔ ∀

L

up

L

:

L

[

f

]

for every mult

f

.

Definition

1833

.

h

f

iX

=

d

X

up

X

h

f

i

X

for every mult

f

whose base is a

complete lattice.

Definition

1834

.

Let

f

be a mult whose base is a complete lattice.

Upgrading

of this mult is square mult

f

with base

f

= core

f

= base

f

and

h

f

i

X

=

h

f

iX

for every

X ∈

Q

base

f

.

Lemma

1835

.

L

i

6 h

f

i

L|

(dom

L

)

\{

i

}

⇔ ∀

L

up

L

:

L

i

6 h

f

i

L

|

(dom

L

)

\{

i

}

,

if every ((base

f

)

i

,

(core

f

)

i

) is a primary filtrator over a meet-semilattice with least

element.