background image

Multireloids Relationships

by Victor Porton

Web:

http://www.mathematics21.org

December 12, 2014

Fix an indexed family

U

of sets.

Denition 1.

Let

f

be a multireloid of the form

U

.

h

(

FCD

)

f

i

n

X

=

d

F

2

up

f

F

h

"

F

i

n

X

for every

n

2

dom

U

and indexed family

X 2

(

i

2

(

dom

U

)

n f

n

g

:

F

(

U

i

))

.

Theorem 2.

L 2

[(

FCD

)

f

]

,8

F

2

up

f

:

L 2

[

"

F

]

.

Proof.

L

n

/

h

(

FCD

)

f

i

n

Lj

(

dom

U

)

nf

n

g

,L

n

/

d

F

2

up

f

F

h

"

F

i

n

Lj

(

dom

U

)

nf

n

g

,

??

, 8

F

2

up

f

:

L

n

/

h

"

F

i

n

Lj

(

dom

U

)

nf

n

g

,8

F

2

up

f

:

L 2

[

"

F

]

.

Corollary 3.

(

FCD

)

f

is a multifuncoid of the form

i

2

dom

U

:

F

(

U

i

)

.

Proof.

Thus

L

n

/

h

(

FCD

)

f

i

n

Lj

(

dom

U

)

nf

n

g

,L

m

/

h

(

FCD

)

f

i

m

Lj

(

dom

U

)

nf

m

g

.

So

(

FCD

)

f

is a pre-multifuncoid. That it is an upper set is obvious.

Theorem 4.

(

FCD

)

f

is always a completary staroid.

Proof.

L

0

t L

1

2

[(

FCD

)

f

]

,8

F

2

up

f

:

L

0

t L

1

2

[

"

F

]

,

(CONJECTURE 17.71)

1