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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).

1

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.

2

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.

3

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.

4

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

).

5

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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