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Upgrading a Multifuncoid

∗

May 1, 2012

Abstract

I deﬁne the concepts of multifuncoid (and completary multifun-

coid) and upgrading. Then I conjecture that upgrading of certain mul-

tifuncoids are multifuncoids (and that upgrading certain completary mul-

tifuncoids are completary multifuncoids). I have proved the conjectures

for n 6 2.

The main conjecture from this article is now p roved in the article

“Multidimensional Funcoids“

This short article is the ﬁrst my public writing wher e I introduce the concept

of multidimensional funcoid which I am investigating now.

Refer to this Web site for the theory which I now attempt to generalize.

1 Background

1.1 About some p osets

Let A is a poset that is a set partially ordered by a relation ≥.

If A is a join-semilattice, I will denote a ⊔ b join o f its elements. (Dually for

a meet-semilattice I will de note a ⊓ b meet of its elements.)

If A = A

i∈n

is a family o f posets, then I will denote

Q

A the product order

on A that is we have for every a, b ∈

Q

A

a ≥ b ⇔ ∀i ∈ n : a

i

≥ b

i

.

Note that if every A

i

is a join-semilattice then

Q

A is als o a join-semilattice

and

a ⊔ b = λi ∈ n : a

i

⊔ b

i

.

I will denote A

n

=

Q

i∈n

A for every pose t A and an index set n.

∗

Keywords: multifuncoid, ﬁltrator; A.M.S. subject classiﬁcation: 54J05, 54A99,

54B99, 54D35, 54D70, 54E05

1

1.2 Filtrators and upgrading

Deﬁnition 1 A ﬁltrator is a pair (A; Z) of a poset A and its subset Z.

See [2] for a detailed study of ﬁltrators .

Having ﬁxed a ﬁltrator, we deﬁne:

Deﬁnition 2 up x = {Y ∈ Z | Y > x} for every X ∈ A.

Deﬁnition 3 E

∗

K = {L ∈ A | up L ⊆ K} (upgrading the set K) for every

K ∈ PZ.

1.3 Multifuncoids

Deﬁnition 4 A free star on a join-semilattice A with least element 0 is a set

S such that 0 6∈ S and

∀A, B ∈ A : (A ⊔ B ∈ S ⇔ A ∈ S ∨ B ∈ S) .

I will denote the set of free stars on A as A Star.

Let n be a s e t. As an example, n may be an ordinal, n may be a natural

number, considered as a set by the formula n = {0, . . . , n − 1}. Let A = A

i∈n

is a family of posets index ed by the set n.

Deﬁnition 5 Let f ∈ P

Q

A, i ∈ dom A, L ∈

Q

A|

(dom A)\{i}

.

(val f )

i

L = {X ∈ A

i

| L ∪ {(i; X)} ∈ f} .

(“val” is an abbreviation of the word “value”.)

Proposition 1 f can be restored knowing (val f)

i

for some i ∈ n.

Proof f = {K ∈

Q

A | K ∈ f } =

L ∪ {(i; X)} | L ∈

Q

A|

(dom A)\{i}

, X ∈ A

i

, L ∪ {(i; X)} ∈ f

=

L ∪ {(i; X)} | L ∈

Q

A|

(dom A)\{i}

, X ∈ (val f)

i

L

.

Deﬁnition 6 Let A is a family of join-semilattices. A pre-multidimensional

funcoid (or pre-multifuncoi d for short) of the form A is an f ∈ P

Q

A

such that we have that: (val f )

i

L is a free star for every i ∈ dom A, L ∈

Q

A|

(dom A)\{i}

.

Deﬁnition 7 A multidimensional funcoid ( or multifuncoid for short) is

a pre-multifuncoid which is an upper set.

Proposition 2 If L ∈

Q

A and L

i

= 0

A

i

for some i then L 6∈ f if f is a

pre-mu ltifuncoid.

Proof Let K = L|

dom A\{i}

. We have 0 6∈ (val f)

i

K; K ∪ {(i; 0 )} 6∈ f ; L 6∈ f .

2

Deﬁnition 8 Inﬁnitary pre-multifuncoid is such an n-ary multifuncoid that

n is inﬁnite; ﬁnitary pre-mul tifuncoid is such an n-ary multifuncoid that n

is ﬁnite.

Deﬁnition 9 Let A is a family of join-semilattices. A completary multi fun-

coid of the form A is an f ∈ P

Q

i∈dom A

A

i

such that

1. L

0

⊔ L

1

∈ f ⇔ ∃c ∈ {0, 1}

n

:

λi ∈ n : L

c(i)

i

∈ f for every L

0

, L

1

∈

Q

A.

2. If L ∈

Q

A and L

i

= 0

A

i

for some i then ¬f L.

Proposition 3 A completary multifuncoid is a multifuncoid.

Proof Let f is a completary multifuncoid.

Let K ∈

Q

i∈(dom A)\{i}

A

i

. Let L

0

= K ∪ {(i; X

0

)}, L

1

= K ∪ {(i; X

1

)}

for some X

0

, X

1

∈ A

i

. Then X

0

⊔ X

1

∈ (val f )

i

K ⇔ L

0

⊔ L

1

∈ f ⇔ ∃k ∈

{0, 1} : K ∪ {(i; X

k

)} ∈ f ⇔ K ∪ {(i; X

0

)} ∈ f ∨ K ∪ {(i; X

1

)} ∈ f ⇔ X

0

∈

(val f )

i

K ∨ X

1

∈ (val f )

i

K.

So (val f)

i

K is a free star (taken in account that K

i

= 0

A

i

⇒ f 6∈ K).

It remained to prove that f is an upper set. Let L

0

≤ L

1

for some L

0

, L

1

∈

Q

A and L

0

∈ f . Then taking c = n × { 0} we get λi ∈ n : L

c(i)

i = λi ∈ n :

L

0

i = L

0

∈ f and thus L

1

= L

0

⊔ L

1

∈ f .

Proposition 4 Every ﬁnitary pre-multifuncoid is completary.

Proof ∃c ∈ {0, 1}

n

: (λi ∈ n : L

c(i)

i) ∈ f ⇔

∃c ∈ {0, 1}

n−1

: ({(n − 1; L

0

(n − 1))} ∪ {(i; L

c(i)

i) | i ∈ n − 1} ∈ f ∨

{(n − 1; L

1

(n − 1))} ∪ { (i; L

c(i)

i) | i ∈ n − 1} ∈ f) ⇔ ∃c ∈ {0, 1}

n−1

:

{(n − 1; L

0

(n − 1) ⊔ L

1

(n − 1))} ∪ {(i; L

c(i)

i) | i ∈ n − 1} ∈ f ⇔ . . . ⇔

{(i; L

0

i ⊔ L

1

i) | i ∈ n} ∈ f .

Theorem 1 For ﬁnite n the following are the same:

1. pre-mu ltifuncoids;

2. multifuncoids;

3. completary multifuncoids.

Proof f is a ﬁnitary pre-multifuncoid ⇒ f is a ﬁnitary completary multifun-

coid.

f is a ﬁnitary completary multifuncoid ⇒ f is a ﬁnitary multifuncoid.

f is a ﬁnitary multifuncoid ⇒ f is a ﬁnitary pre-multifuncoid.

As it will b e clear from below, (ﬁnitary) multifuncoids are a generalization

of funco ids [1].

I will deno te AFCD the se t of multifuncoids for a ﬁnite family A of join-

semilattices.

3

2 Open problems

Conjecture 1 Let ℧ be a set , F be the set of ﬁlters on ℧ ordered reverse to

set-theoretic inclusion, P be the set of principal ﬁlters on ℧, let n be an index

set. Consider the ﬁltrator (F

n

; P

n

). If f is a multifuncoid of the form P

n

, then

E

∗

f is a multifuncoid of the form F

n

.

A similar conjecture about completary multifuncoids:

Conjecture 2 Let ℧ be a set , F be the set of ﬁlters on ℧ ordered reverse to

set-theoretic inclusion, P be the set of principal ﬁlters on ℧, let n be an index

set. Consider the ﬁltrator (F

n

; P

n

). If f is a completary multifuncoid of the

form P

n

, then E

∗

f is a completary mult ifuncoid of the form F

n

.

A weaker conjecture:

Conjecture 3 Let ℧ be a set , F be the set of ﬁlters on ℧ ordered reverse to

set-theoretic inclusion, P be the set of principal ﬁlters on ℧, let n be an index

set. Consider the ﬁltrator (F

n

; P

n

). If f is a completary multifuncoid of the

form P

n

, then E

∗

f is a multifuncoid of the form F

n

.

For ﬁnite n all three conjectures are equivalent.

For n = 0 the conjecture is trivial. For n = 1 it can be proved using the

theory of ﬁlters [2]. For n = 2 we can prove it using the theory of funcoids [1].

For card n > 3 (ﬁnite and inﬁnite) the problem is open.

The full proo fs for c ard n 6 2 are presented below.

If a conjecture will be proved true, we may generalize it for a wider set of

ﬁltrators.

3 Preliminary Results

3.1 Isomorphic ﬁlt rators

We will use the concept of isomorphic ﬁltrators in the below proo fs.

An isom orphism from a ﬁltrator (A

0

; Z

0

) to a ﬁltrator (A

1

; Z

1

) is an order

embedding ϕ from A

0

to A

1

such that the image of Z

0

under ϕ is exactly Z

1

.

Two ﬁltrators are isomorphic when there exist an isomorphism fr om one to

the other.

It is triv ial that neither the prope rty of be ing a multifuncoid, nor the result

of upgrading does change under an isomorphism.

3.2 The proof of the conjecture for card n 6 2

The below constitutes a proof of my conjecture for n ∈ {0, 1, 2} as well as n

being any set of cardinality card n 6 2, because a particular index se t doe s not

matter, just it’s cardinality.

Let Z = P

n

and A = F

n

.

4

3.2.1 The proof for n = 0

In this case a multifuncoid f of the form Z = P

0

= {()} is an element of the set

PB

0

= {{()}} that is f = {()}. Obviously f is an upper set. Then

E

∗

f =

L ∈ F

0

| up L ⊆ f

= {() | up () ⊆ {()}} = {() | {()} ⊆ {( )}} = {()} .

For i = dom F

0

we have (val f)

i

L is a free star just because i doesn’t exist.

Obviously E

∗

f is a n upper set.

So E

∗

f is a multifuncoid of the form F

0

.

3.2.2 The proof for n = 1

We will use notation from [2].

Lemma 1 The upgrading (regarding the ﬁltrator (F; P)) of every free star on

P is a free star on F.

Proof Let f is a free star on P. Then (theorem 45 in [2]) there exist a g ∈ F

such that ∂g = f .

E

∗

f = {L ∈ F | up L ⊆ ∂g} = {L ∈ F | L 6≍ g}. It remained to prove

that {L ∈ F | L 6≍ g} is a free star.

Obviously 0

F

6∈ {L ∈ F | L 6≍ g}.

For every A, B ∈ {L ∈ F | L 6≍ g} we have A⊔B ∈ {L ∈ F | L 6≍ g} ⇔

(A ⊔ B) ⊓ g 6= 0

F

⇔ (A ⊓ g) ⊔ (B ⊓ g) 6= 0

F

⇔ A ⊓ g 6= 0

F

∨ B ⊓ g 6= 0

F

⇔ A ∈

{L ∈ F | L 6≍ g} ∨ B ∈ {L ∈ F | L 6≍ g}.

The proof is ﬁnished.

Let Q be the set of all multifuncoids of the form A

1

where A is a join-

semilattice with least element. Then f ∈ Q if and only if (val f )

0

∅ is a free

star.

But (val f )

0

∅ = {X ∈ A | {(0; X)} ∈ f } = {X ∈ A | f0 = X} = f 0.

So it uis easy to show that the ﬁltrator of the form

F

1

FCD; P

1

FCD

is

isomorphic to the ﬁltrator (F Star; P Star).

Thus by the lemma upgrading a multifuncoid of the form P

1

is a multifuncoid

of the form F

1

.

3.2.3 The proof for n = 2

An f is a (ﬁnitary) multifuncoid of the form A × B (for A, B being join-

semilattices with least elements) iﬀ all the following:

1. (val f )

0

L is a free star for every L = {(1; Y )} where Y ∈ B;

2. (val f )

1

L is a free star for every L = {(0; X)} where X ∈ A;

what is equal to the following:

1. {X ∈ A | X f Y } is a free star for e very Y ∈ B;

5

2. {Y ∈ B | X f Y } is a free star for every X ∈ A;

what is equal to the following:

1. (I ⊔ J) f Y ⇔ I f Y ∨ J f Y and not 0 f Y for every Y ∈ B, I, J ∈ A;

2. X f (I ⊔ J) ⇔ X f I ∨ X f J and not X f 0 for every X ∈ A, I, J ∈ B.

By the way, it implies that f 7→ [f ]

∗

is a bijection from the set of funcoids from

℧

0

to ℧

1

into the set of multifuncoids of the form P℧

0

× P℧

1

, for every sets

℧

0

and ℧

1

.

Now supp ose f is a multifuncoid of the form P

2

. Then:

1. (I ⊔ J) f Y ⇔ I f Y ∨ J f Y and not 0 f Y for every Y, I, J ∈ P;

2. X f (I ⊔ J) ⇔ X f I ∨ X f J and not X f 0. for every X, I, J ∈ P.

Thus multifuncoids of the fo rm P

2

are essentially equivalent to funcoids from

P to P ([1]), formally: there exist a funcoid f

′

such that [f

′

]

∗

= f .

E

∗

f =

L ∈ F

2

| up L ⊆ f

= {(L

0

; L

1

) | L

0

, L

1

∈ F, ∀g

0

∈ up L

0

, g

1

∈ up L

1

: (g

0

; g

1

) ∈ f } =

(L

0

; L

1

) | L

0

, L

1

∈ F, ∀g

0

∈ up L

0

, g

1

∈ up L

1

: g

0

[f

′

]

∗

g

1

= {(L

0

; L

1

) | L

0

, L

1

∈ F, L

0

[f

′

] L

1

} =

[f

′

].

Thus:

1. (I ⊔ J) (E

∗

f) Y ⇔ I (E

∗

f) Y ∨ J (E

∗

f) Y and not 0 (E

∗

f) Y for every

Y, I, J ∈ F;

2. X (E

∗

f) (I ⊔ J) ⇔ X (E

∗

f) I ∨ X (E

∗

f) J and not X (E

∗

f) 0 for every

X, I, J ∈ F.

that is E

∗

f is a multifuncoid of the form F

2

.

References

[1] Victor Porton. Funcoids and reloids. At

http://www.mathematics21.org/binaries/funcoids-reloids.pdf.

[2] Victor Porton. Filters on posets and generalizations. International Journal

of Pure and Applied Mathematics, 74(1):55–119, 2012.

6