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

Remark 1. It is a very rough partial draft. It is meant to express a rough research idea, not to
http://www.mathematics21.o rg/algebraic-gene ral-topology.html upon which this formalistic is
based (especially about the deﬁnition of generalized limit).
The idea is simple (for these who know funcoids theory). But to nd exact formulations abo ut thi s
is notoriously diﬃcult. Below are attempts to formulate things about the theory of singularit ie s.
New theory
Deﬁnition 2. Singularity level is a transitive, T
2
-separable endofuncoid.
Let ν be a singularity level. Let be a ﬁlter.
Deﬁne SLA(ν) as:
Ob SLA(ν) = {ν f | f is a monovalued funcoid with domain }
X [SLA(ν)]
Y x X K GR xL Y : L K
[FIXME: It is probably not a funcoid .]
Remark 3. GR x is used despite of it is a funcoid not reloid.
Proposition 4. SLA(ν) is an endofuncoid.
Proof. ¬( [SLA(ν)]
Y ) and ¬(X [SLA (ν)]
) are obvious.
I J [SLA(ν)]
Y I [SLA(ν)]
Y J [SLA(ν)]
Y is obvious.
X [SLA(ν)]
I J x X K GR x L I J: L K x X K GR x :
(L I: L K L J: L K)??
??
Alternative deﬁnition:
[FIXME: It is probably not a funco id.]
Deﬁnition 5. X [SLA(ν)]
Y z Ob µK GR z x X , y Y : x, y K
Proposition 6. SLA(ν) is a funcoid.
Proof. X [SLA(ν)]
Y z Ob µ , x X, y Y K (GR z)
X ×Y
: x, y K
x,y
??
I J [SLA(ν)]
Y z Ob µK GR z x I J , y Y : x, y K z Ob µ K GR z:
( y Y : y K (x I: x K x J : x K)) z Ob µK GR z: (( y Y : y K x I:
x K) ( y Y : y K x J: x K)) ??
Proposition 7. SLA(ν) is T
2
-separable.
Proof. ??
Proposition 8. SLA(ν) is transitive.
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Proof. ??
Galufuncoids
Let A and B are Rel-morphisms. I will denote like (
A
) = GR A and (
B
) = GR B.
Deﬁnition 9. Galufuncoi ds between A and B is a quadruple (A; B; α; β) such that
x Ob A, y Ob B: (αx
B
y x
A
βy).
Deﬁnition 10. x [f] y x
Src f
βy.
Obvious 11. x [f ] y x
Src f
βy αx
Dst f
y.
Remark 12. Galuf uncoids are a gener alization of both (pointfree) funcoids and Galois connec -
tions.
Deﬁnition 13. The reverse galufuncoid is deﬁned by the formula:
(A; B; α; β)
1
= (B; A; β; α).
Proposition 14. Composition of (composable) galufuncoids is a galufuncoid.
Proof. (α
2
α
1
)x y α
2
α
1
x y α
1
x β
2
y x β
1
β
2
y x (β
1
β
2
)y.
Obvious 15. Galufuncoids form a category (sim ilarly to the category of pointfree funcoids).
Deﬁnition 16. On the set of galufuncoids is deﬁned a preo rder by the formula: f g [f ][g].
Galufuncoidal product
Functional galufuncoid
Deﬁnition 17. Functional galufuncoid ν/∆ of ν thr ough ﬁlter is the endo-galufuncoid deﬁned
by the formulas:
Ob(ν/∆) = FCD(Base(∆); Ob ν)
hν/∆if = ν f and h(ν/∆)
1
if = ν
1
f
f
Ob(ν/∆)
g g
1
f id
FCD
[TODO: Restric t to the special case f = ν F to make i t T
2
.]
[TODO: X [SLA(f )] Y is deﬁned as existence of x X such that for every entourage of x ther e is
y Y which is a subﬁlter of this entourage.]
Obvious 18.
Ob(ν/∆)
is a symmetric relatio n.
Proposition 19. This is really a galufuncoid and f [ν/∆] g g
1
ν f id
.
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Proof. We need to prove
hν/∆if
Ob(ν/∆)
g g
1
ν f id
f
Ob(ν/∆)
h(ν/∆)
1
ig .
Really,
hν/∆if
Ob(ν/∆)
g ν f
Ob(ν/∆)
g g
1
ν f id
f
Ob(ν/∆)
h(ν/∆)
1
ig f
Ob(ν/∆)
ν
1
g g
1
ν f id
Remark 20. A way to come to the above formula
x atoms ∆: fx [ν] gx x atoms : x [g
1
ν f ] x g
1
ν f id
.
Hierarchy of singularities
Consider two endo-galufuncoids µ and ν. Values on Ob µ will be have like arguments of functions,
of Ob ν like values of functions.
I call SLA(Ob µ) singularity level above Ob µ the set of sets of funcoids ν f |
hµi
{x}
(or alternatively
of lim its xlim f |
hµi
{x}
) where f is a monovalued principal funcoid in FCD(Ob µ; Ob ν).
[TODO:
Maybe exclude the zero funcoid?]
Consider a galufuncoid ω deﬁned by the formulas:
hωif = ν f and hω
1
if = f ν and f [ω] g x Ob ν: g
1
f id
hµi
{x}
FCD
.
We need to prove
hωix
ω
g
1
x Ob ν: g
1
(
ν
) f id
hµi
{x}
and ??
The ﬁrst is equivalent to (
ν
) f
ω
g
1
x Ob ν: g
1
(
ν
) f id
hµi
{x}
FCD
.
Really, (
ν
) f
ω
g
1
x Ob ν: g
1
(
ν
) f id
hµi
{x}
Lemma 21. Let ν ν ν and ν
1
ν ν. If x and y are ultraﬁ lters, then x [ν] y hν ix = hν iy.
Proof.
[TODO: Prove fo r the more general case of galufuncoids. It’s problematic.]
x [ν] y y hν ix hν iy hν ν ix hν iy hν ix. So taking symmetry into account we have
x [ν] y hν ix = hν iy.
Let now hν ix = hν iy, Then y
hν
1
ν ix; y hν
1
ν ix; y hν ix; y
hν ix; x [ν] y.
Theorem 22. f [ν/hµi
{x}] g ν f |
hµi
{x}
=ν g |
hµi
{x}
for f , g SLA(Ob µ).
[TODO:
Generalize it for any instead of hµi
{x}?]
Proof. It’s enough to prove hf ix [ν] hgix hν ihf ix = hν ihgix for every ultraﬁlter x.
hf ix [ν] hg ix hgix
Ob µ
hν ihf ix ?? hν ihgix
Ob µ
hν ihf i
??
Theorem 23. SLA(Ob µ) is T
2
-separable .
The reloid ν
on the set SLA(Ob ν) could be deﬁned by one of the two formulas below:
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For every point x get
σ
f
(x) =
u
[
f | dom u = hµi
{x}
.
q
f
=
F
xOb µ
σ
f
(x).
f [ν
] g x Ob µ:
G
q
f
[ν/hµi
{x}]
G
q
g
.
or
f [ν
] g ν q
f
= ν q
g
.
[TODO: How to deﬁne galufuncoids corresponding to the above formulas (if at all possible)?
x D: f [ν(x)] g x D: g
hν(x)if ?? g
F
xD
ν(x).
The ?? does not generally hold: our lattices are co-brouwerian not brouwerian!
]
The above formula holds if g is a discrete reloids. So repla ce every funcoid f SLA(Ob ν) with
(RLD)
in
f. Then continue for arbitrary reloids.
Another try: hν
ix = ν
FS
x
y
hν
ix y
ν
FS
x ν
y (
FS
x)
1
ν
1
(
FS
x) y
1
[FIXME: x and y are of
diﬀerent types.]
x
hν
1
iy x
ν
1
FS
y
The rest
One more ot he r deﬁnition:
f [ν
′′
] g
G[
g
1
ν
G[
f
0
Yahoo! (i j) [ν
′′
] g i [ν
′′
] g j [ν
′′
] g etc.
Proof. (i j) [ν
′′
] g (
FS
g)
1
ν (
FS
(i j))
0 (
FS
g)
1
ν ((
FS
i) (
FS
j))
0
(
F S
g)
1
ν (
F S
i) (
F S
g)
1
ν (
F S
j)
0 (
F S
g)
1
ν (
F S
i)
0
(
FS
g)
1
ν (
FS
j)
0 i [ν
′′
] g j [ν
′′
] g
Proposition 24. ν
′′
is a galufuncoid.
Proof. ??
An attempt of an alternate deﬁnition:
f [ν
] g xlim f ν xlim g
0 [FIXME: Does this make sense?] [TODO: Diﬀere ntiate generalized
limit as a set of funcoids or its variation as a funcoid-value function.]
Proposition 25. xlim f SLA(Ob ν) if f FCD(Ob µ; O b ν) \
0
FCD(Ob µ;Ob ν)
.
Proof. ??
Proposition 26. τ (x) SLA(O b ν).
Proof. ??
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Proposition 27. τ (x) [ν
] τ (y) x [ν] y.
Proof. ??
Metasingular numbers
Let y SLA(Ob µ). I will denote r(y) such x Ob ν that τ (x) = y, if such x exis ts.
I will call base singular numbers the set BSN = Ob ν SLA(Ob ν) SLA(SLA(Ob ν))
.
[TODO: Set that this union is di sjoint.]
I will call meta-singular numbers the set MSN = {y SLA(Ob ν) | x BSN: y = τ (x)}.
Deﬁnition 28. I call reduced BSN its corresponding MSN (that is r applied to our BSN a natural
number of times while possible).
Deﬁnition 29. I call reduced limit the reduced generalized limit.
Functions with meta-sin gular numbers as arguments
Let f is an n-ary (n is an arbitrary possibly inﬁnite index set) function o n O b ν. Then deﬁne
function f
on SLA(Ob ν) as:
f
(b) =
(
g
Y
(A)
b | g f
)
.
We c an’t use cross-composition product instead of above sub-atomic product because cross -compo-
sition product is not a funcoid (just a pointfree fun coid). We can replace sub-atomic product with
displaced product, but as about my opinion displaced product seems more weird an inconvenient.
The above ind uc e s a trivial deﬁnition of functions on MSN but only for functions of ﬁnite arity
(because having a ﬁnite set of MSN we can raise them to the same (maximum) level).
On diﬀerential equations
Replacing limit in the deﬁnition of derivative with the a bove deﬁned reduced limit, the base set
Ob µ with MSN and operations f on the set Ob µ with corresponding operations on MSN, we get
a new interpretation of a diﬀerential equation (DE) (ordinary or partial).
Let call such (enhance d) diﬀerential e quations m eta-singular equations (as opposed to non-singular
equations that is customary diﬀerential equations).
There arise the following questions:
Deﬁnition 30. I call a solution of a DE a trivia l restriction if it is a restr ic tio n (to t he set of non-
singular points) of exactly one enhanced DE.
We need to ﬁnd when there are solutions which are not trivial restrictions.
Then we can split such non-trivial solutions into follow ing classes:
“added solutions” are solutions whose restriction t o non-singularity points is not a non-
singular solution;
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“alternate solutions” is when an non-singular solution is a restriction of more than one meta-
singular solution;
“disappearing solutions” when a non-singular solution is not a restriction of a meta-singular
solution.
Special case of general relativity
I am not a expert in general relativity (I am not even a professional mathematician).
But it looks like that the equations of general relativity can be converted (as described above) into
meta-singular equations. For the special case of general relativity equa tions, the above classes are:
“added solutions” would possibly characterize a “world above” described not with real num-
bers as our world but with singularities. This may or may not be of physical interest.
“alternate solutions” would characterize black (or white) holes with additional information
hidden inside. This a dditional information may probably solve the well k nown pa radox of
information disappearing when it fa lls into a black hole.
“disappearing solutions” would mean that the laws of nature are possibly more restrictive
than c onsidered in more traditional physics. Could it resolve time-machine related p ara-
doxes?
I again repeat that I am not an expert in general relativity. I seek collaboration w ith general
relativity experts to solve the problems I’ve formulated.
I think (except of the case of the negative result that is there are no non-trivial solutions) this
research is destined to receive Nobe l Prize and/or Fundamental Physics Prize. I want my half.
Note that the g roup G (see the deﬁnition of gene ralized limit in my book) for general relativity
can be deﬁned in two diﬀerent ways: as the group of homeomo rpisms of the curved space or as the
group of only uniformly continuous (in both directions) bijections. This gives us two new theories
of general relativity.
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