Algebraic General Topology. Vol 1: Paperback / E-book || Axiomatic Theory of Formulas: Paperback / E-book

May 1, 2012
Abstract
I deﬁne the concepts of multifuncoid (and completary multifun-
tifuncoids are multifuncoids (and that upgrading certain completary mul-
tifuncoids are completary multifuncoids). I have proved the conjectures
for n 6 2.
“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
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
in
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
in
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
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
in
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
idom 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}
n1
: ({(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}
n1
:
{(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 AB {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