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

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Open Problems in Alg ebraic General Topology
by Victor Porton
September 10, 2016
Abstract
This document lists in one place all conjectures and open problems in my Algebraic General
Topology research which were yet not solved. This document also contains other relevant
materials such as proved theorems related with the conjectures.
Table of contents
Organizational info . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Misc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Provability without axiom of choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Complete funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Relationships of funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Connectedness of funcoids and reloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Algebraic properties of S and S
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Oblique products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Compactness and Heine-Cantor theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Category theory related . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Organizational info
Discuss these problems and their solutions in the algebraic-general-topology Google group.
See http://www.mathematics21.org/algebraic-general-topology.html for more details.
See also http://filters.wikidot.com/open-problems for open problems about filters.
Read http://www.mathematics21.org/solvers.html if you solved any of the below problems in order
that I could nominate you for Abel Prize if I found your solutions worth it.
Misc
Conjecture 1. A reloid f is monovalued iff
8g 2 RLD(Src f; Dst f): (g v f ) 9A 2 F(Src f ): g = f j
A
):
. This document has been written using the GNU T
E
X
MACS
text editor (see www.texmacs.org).
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Conjecture 2. The filtrator of funcoids is:
1. with separable core;
2. with co-separable core.
Conjecture 3. Let f be a set, F be the set of f.o. on f, P be the set of principal f.o. on f, let
n be an index set. Consider the filtrator (F
n
; P
n
). Then if f is a completary multifuncoid of the
form P
n
, then f is a completary multifuncoid of the form F
n
.
Conjecture 4. Let f
1
and f
2
are monovalued, entirely defined funcoids with Srcf
1
= Srcf
2
= A.
Then there exists a pointfree funcoid f
1
×
(D)
f
2
such that (for every fitler x on A)
f
1
×
(D)
f
2
x =
G
fhf
1
i X ×
FCD
hf
2
i X j X 2 atoms xg:
(The join operation is taken on the lattice of filters with reversed order.)
A positive solution of this problem may open a way to prove that some funcoids-related categories
are cartesian closed.
Conjecture 5. b /
Anch(A)
StarComp(a; f ) , 8A 2 GR a; B 2 GR b; i 2 n: A
i
[f
i
] B
i
for anchored
relations a and b on powersets.
It's consequence:
Conjecture 6. b /
Anch(A)
StarComp(a; f) , a /
Anch(A)
StarComp(b; f
y
) for anchored relations a
and b on powersets.
Conjecture 7. b /
Strd(A)
StarComp(a; f ) , a /
Strd(A)
StarComp(b; f
y
) for pre-staroids a and b on
powersets.
Conjecture 8. f v
Q
RLD
a, 8i 2 arity f: Pr
i
RLD
f v a
i
for every multireloid f and a
i
2F((form f )
i
)
for every i 2 arity f .
Conjecture 9. L 2 [f ])[f]\
Q
i2 dom A
atomsL
i
=/ ; for every pre-multifuncoid f of the form whose
elements are atomic posets. (Does this conjecture hold for the special case of form whose elements
are posets on filters on a set?)
Conjecture 10. The formula f t
FCD(A)
g 2 cFCD(A) is not true in general for completary
multifuncoids (even for multifuncoids on powersets) f and g of the same form A.
Conjecture 11. GR StarComp(a t
pFCD
b; f) = GR StarComp(a; f) t
pFCD
GR StarComp(b; f) if f
is a pointfree funcoid and a, b are multifuncoids of the same form, composable with f .
Conjecture 12. Every metamonovalued funcoid is monovalued.
Conjecture 13. Every metamonovalued reloid is monovalued.
Conjecture 14. Every monovalued reloid is metamonovalued.
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Problem 15. Let A and B be infinite sets. Characterize the set of all coatoms of the lattice
FCD(A; B) of funcoids from A to B. Particularly, is this set empty? Is FCD(A; B) a coatomic
lattice? coatomistic lattice?
Hyperfuncoids
Let A be an indexed family of sets.
Products are
Q
A for A 2
Q
A.
Hyperfuncoids are filters F¡ on the lattice ¡ of all finite unions of products.
Problem 16. 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¡.
Provability without axiom of choice
Conjecture 17. Distributivity of the lattice FCD(A; B) of funcoids (for arbitrary sets A and B)
is not provable in ZF (without axiom of choice).
Conjecture 18. a n
b = a#b for arbitrary filters a, b on powersets is not provable in ZF (without
axiom of choice).
Complete funcoids and reloids
Question 19. Is ComplFCD(A; B) a co-brouwerian lattice?
Conjecture 20. Composition of complete reloids is complete.
Conjecture 21. Compl f u Compl g = Compl(f u g) for every reloids f ; g 2 RLD(A; B) (for every
sets A, B).
Conjecture 22. Compl f = f n
(Ω(Src f) ×
FCD
1
F(Dst f )
) for every funcoid f.
Conjecture 23. Compl f = f n
(Ω(Src f) ×
RLD
1
F(Dst f )
) for every reloid f .
Question 24. Is ComplRLD(A; B) a distributive lattice? Is ComplRLD(A; B) a co-brouwerian
lattice? (for every sets A and B).
Conjecture 25. Let A, B, C be sets. If f 2 RLD(B; C) is a complete reloid and R 2 PRLD(A; B)
then f
F
R =
F
hf iR.
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Conjecture 26. Every entirely defined monovalued isomorphism of the category of funcoids is a
discrete funcoid.
Conjecture 27. For composable reloids f and g it holds
1. Compl(g f) = (Compl g) f if f is a co-complete reloid;
2. CoCompl(f g) = f CoCompl g if f is a complete reloid;
3. CoCompl((Compl g) f) = Compl(g (CoCompl f)) = (Compl g ) (CoCompl f );
4. Compl(g (Compl f)) = Compl(g f);
5. CoCompl((CoCompl g) f) = CoCompl(g f ).
Relationships of funcoids and reloids
Conjecture 28. (RLD)
¡
f = (RLD)
in
f for every funcoid f .
Conjecture 29. (RLD)
in
(g f) = (RLD)
in
g (RLD)
in
f for every composable funcoids f and g.
Conjecture 30. (RLD)
out
id
A
FCD
= id
A
RLD
for every filter A.
Conjecture 31. (RLD)
in
is not a lower adjoint (in general).
Conjecture 32. (RLD)
out
is neither a lower adjoint nor an upper adjoint (in general).
Conjecture 33. If A ×
RLD
B v (RLD )
in
f then A ×
FCD
B v f for every funcoid f and A 2 F(Src f ),
B 2 F(Dst f).
Conjecture 34. ρ
d
F =
d
hρiF for a set F of reloids. (ρ is defined in [1])
Conjecture 35. For every funcoid g
1. Cor (RLD)
in
g = (RLD)
in
Cor g;
2. Cor (RLD)
out
g = (RLD)
out
Cor g.
Conjecture 36. For every composable funcoids f and g
(RLD)
out
(g f ) w (RLD)
out
g (RLD)
out
f:
Connectedness of funcoids and reloids
Conjecture 37. A filter A is connected regarding a funcoid µ iff A is connected for every discrete
funcoid F 2 up µ.
Conjecture 38. A filter A is connected regarding a reloid f iff it is connected regarding the
funcoid (FCD)f.
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Conjecture 39. Let A is a filter and F is a binary relation on A × B for some sets A, B.
A is connected regarding "
FCD(A;B)
F iff A is connected regarding "
RLD(A;B)
F .
Proposition 40. The following statements are equivalent for every endofuncoid µ and a set U:
1. U is connected regarding µ.
2. For every a; b 2 U there exists a totally ordered set P U such that min P = a, max P = b,
and for every partion fX ; Y g of P into two sets X, Y such that 8x 2 X ; y 2 Y : x < y, we
have X [µ]
Y .
Algebraic properties of S and S
Conjecture 41. S(S(f )) = S(f) for
1. any endo-reloid f;
2. any endo-funcoid f.
Conjecture 42. For any endo-reloid f
1. S(f ) S(f ) = S(f);
2. S
(f ) S
(f) = S
(f );
3. S(f ) S
(f ) = S
(f) S(f ) = S
(f).
Conjecture 43. S(f ) S(f ) = S(f ) for any endo-funcoid f .
Oblique products
Conjecture 44. A ×
F
RLD
B @ A n B for some f.o. A, B.
A stronger conjecture:
Conjecture 45. A ×
F
RLD
B @ A n B @ A ×
RLD
B for some f.o. A, B. Particularly, is this formula
true for A = B = \ "
R
(0; +1)?
Products
Conjecture 46. Cross-composition product (for small indexed families of reloids) is a quasi-
cartesian function (with injective aggregation) from the quasi-cartesian situation S
0
of reloids to
the quasi-cartesian situation S
1
of pointfree funcoids over posets with least elements.
Remark 47. The above conjecture is unsolved even for product of two multipliers.
Conjecture 48. a
h
Q
(C)
f
i
b , 8i 2 dom f: Pr
i
FCD
a [f
i
] Pr
i
FCD
b for every indexed family f of
funcoids and a 2 FCD(λi 2 dom f : Src f
i
), b 2 FCD(λi 2 dom f : Dst f
i
).
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Conjecture 49. "
FCD
A
h
Q
(C)
f
i
"
FCD
B , "
RLD
A
h
Q
(A)
f
i
"
RLD
B for every indexed family f of
funcoids and a 2 P
Q
i2dom f
Src f
i
, a 2 P
Q
i2dom f
Dst f
i
.
Conjecture 50.
D
Q
(A)
f
E
"
RLD
X = (RLD)
in
D
Q
(C)
f
E
"
FCD
X for every indexed family f of
funcoids and a suitable set X.
Compactness and Heine-Cantor theorem
Theorem 51. Let f be a T
1
-separable compact reflexive symmetric funcoid and g be a reloid such
that
1. (FCD)g = f ;
2. g g
¡1
v g .
Then g = hf × f i
.
About the above conjecture see also
http://www.openproblemgarden.org/op/direct_proof_of_a_theorem_about_compact_funcoids
8F 2 F: (F \ im f =/ ; ) 9α: fαg [f ] F) or equivalently
8F 2 F: ( hf
¡1
iF =/ ; ) 9α: fαg hf
¡1
iF)
is a possible definition of compact funcoid. (A special case of this definition was hinted by Victor
Petrov.) How this is related with open covers and finite covers from the traditional definition of
compactness? Does compactness imply completeness?
Generalize Heine-Cantor theorem for funcoids and reloids.
Category theory related
Conjecture 52. The categories Fcd and Rld are cartesian closed (actually two conjectures).
Bibliography
[1] Victor Porton. Distributivity of compositon with a principal reloid over join of reloids. At http://
www.mathematics21.org/binaries/decomposition.pdf.
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