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Hyperfuncoids

by Victor Porton

Web: http://www.mathematics21.org

December 10, 2014

This is a rough partial draft.

1 Hyperfuncoids

Let A is an indexed family of sets.

Products are

Q

A for A 2

Q

A.

Problem 1. Is

d

FCD

a bijection from hyperfuncoids F¡ to:

1. prestaroids on A;

2. staroids on A ;

3. completary staroids on A?

If yes, is up

¡

deﬁning the inverse bijection?

If not, characterize the image of the function

d

FCD

deﬁned on F¡.

Alternatively (diﬀerently for the inﬁnite dimensional case!) deﬁne ¡ as the set of intersections

of sets with holes that is

Q

A n

Q

A where A v A. In other words, it is the set ¡

∗

of complements

of elements of the set ¡.

Theorem 2. For every anchored relation f on powersets, f =

d

Anch(A)

up

¡

∗

f.

Proof. We need to prove only f v

d

Anch(A)

up

¡

∗

f.

Fix n 2 arity A. Let A 2

Q

(arity f)nfng

A

i

.

[TODO: Deﬁne the complement.]

Deﬁne g(A) =

Q

(arity f)nfng

A

i

× hf i

n

A

[

Q

(arity f)nfng

A

i

× 1

for A 2

Q

(arity f)nfng

A

i

.

Obviously g(A) 2 ¡

∗

.

Let X 2

Q

(arity f)nfng

A

i

.

If 0 =/ X v A then hg(A)i

n

X = hf i

n

A w hf i

n

X.

If X v A then hg(A)i

n

X = 1.

So hG(A)i

n

w hf i

n

and thus G(A) w f .

For a given f, we have hg(A)i

n

A = hf i

n

A. Thus for every A 2

Q

(arity f)nfng

A

i

we have

hf i

n

A v

D

d

cStrd(A)

up

¡

∗

f

E

n

A and so f v

d

Anch(A)

up

¡

∗

f.

Corollary 3.

1. If f is a prestaroid, f =

d

pStrd(A)

up

¡

∗

f.

2. If f is a staroid, f =

d

Strd(A)

up

¡

∗

f.

3. If f is a completary staroid, f =

d

cStrd(A)

up

¡

∗

f.

1