Borromean Objects, as examplified by the group GBorromean Objects, as examplified by the group G168 168

of Klein’s Quartic, linked with Moving Logic of Klein’s Quartic, linked with Moving Logic

CT08 28/06/2008

René Guitart René Guitart Université Paris Diderot Paris 7Université Paris Diderot Paris 7

PLAN

1- Borromean object 1.2- Borromean link 1.3- Aristotle-Appule square and Sesmart-Blanché hexagram 1.4- A « borromean » object in rings: Mat2(GF(2)), in fields : GF(8) 1.5- An observation on Abel.

2- The Klein’s Quartic X(7) and its group G168

3- Moving Logic 3.1. Every function is a (moving) boolean function 3.2. Parametrization of the family of boolean structures 3.3. Logical differentials or cohomological theory of meaning

REFERENCES

[1] F. Klein, Über die Transformationen siebenter Ordnung der elliptischen Funktionen, Math. Ann. 14 (1879), 428-471. Translate in [2]

[2] S. Levy, The Eightfold Way, Cambridge U. Press, 1999.

[3] R. Guitart, Théorie cohomologique du sens, SIC Amiens, 8 novembre 2003, compte-rendu 2004-10/Mars 2004, LAMFA UMR 6140, 39-47.[4] R. Guitart, Moving Logic, from Boole to Galois, Colloque International `Charles Ehrersmann : 100 ans’, 7-9 octobre 2005, Cahiers Top. Géo. Diff. Cat. XVLI-3, 196-198.

1- Borromean Object in a category

B

R

S

I

B/R

B/S

B/I

=

=

=

RI

IS

RS

B = F({r, s, i})/

with: - invariant by the cycle rsir-B/F(r) = F’({s, i})-B/F(s) = F’({i, r})-B/F(i) = F’({r, s})

R = F(r)S = F(s)I = F(i)

Examples : XY = X+Y(of « trivial » binary composition) X Y = 1

1.2-Borromean links

John Robinson’sculptures

In chemistry

Borromeans links

r

s

i

r = reals = symbolici = imaginary

(with a joke in french :rsi = hérésie)

Notations ?

From the psychoanalyst Jacques Lacan:

A clin d’œil To the old Theological use of borromean

Borromeans links : the fundamental group

rir-1sr = *

srs-1is = *

isi-1ri = *

(Computation à la Dehn)

r

s

i

sr is

ri

*

1a

b

c a

b

c

cb-1a

Now, if we add i = 1, the system degeneratesin: sr = sr = sr = * : that is to say : no condition.

1(E3\B)

Z= 1(E3\R)

Z= 1(E3\S)

Z= 1(E3\I)

Z*Z= 1(E3\I)*1(E3\S)

Z*Z= 1(E3\R)*1(E3\S)

Z*Z= 1(E3\R)*1(E3\I)

Associated borromean object in the category of groups

Borromeans links: Tait’s serie

So we get different 1(E3\B),

and so on.

1.3-Aristotle-Appulé square and Sesmat-Blanché hexagone

Necessary Impossible

ContingentPossible

Eventual

Predeterminated

R S

I

R S

R

R S

I

I IS

Associated borromean object in the category of boolean algebras

P(E)

P(R)

P(S)

P(I)

P(I)*P(S)

P(R)*P(S)

P(R)*P(I)

1.4.1- A borromean object in rings: Mat2(GF(2))

r = , s = , i = . 0110

1011

1101

Mat2(GF(2)) is generated by r, s, i freely with the relations:

r2 = s2 = i2 = 1, rs = si = ir , sr = is = ri, r + s + i = 0.

If we do i = 0 we get:r2 = s2 =1, rs = 0 , sr = 0, r + s = 0, So: r = s = 0 = 1.

r+r = s+s =i+i = 0

1.4.2 - Borromean objects in fields: GF(8)

GF(8) = GF(2)[X]/(X3+X2+1)

r, s, i roots of X3+X2+1= 0:

rsi = 1 rs+si+ir = 0 r+s+i = 1

r-1 = i+1 = si s-1 = r+1 = ir i-1 = s+1 = rs

r2 = s s2 = i i2 = r

r+r-1 = s s+s-1 = i i+i-1 = r

GF(8) = {0, 1, r, s, i, r-1, s-1, i-1}

If we do i = 0, we get r = s = i = 0 = 1

1.5- An observation on Abel

1) There is a borromean flavor in Abel approach of elliptics functions:

€

α =dx

(1− c 2x 2)(1+ e2x 2)ϕ 0

ϕ

∫

f = 1− c 2ϕ 2

F = 1+ e2ϕ 2

The three Abel functions , f, F can be normalized in this way:

R = ice, S = ef, I = ieF.

Then we have: R = SI, S = IR, I = RS.

2) The 5th degree problem is related to a « borromean » problem:

x = a+yz, y = b+zx, z = c+xy.

x5-ax4-2x3+(2a-bc)x2 +(1-b2-c2)x-(a+bc) = 0y5-by4-2y3+(2b-ca)y2+(1-c2-a2)y-(b+ca) = 0z5-cz4-2z3 +(2c-ab)z2+(1-a2-b2)z -(c+ab) = 0

Here Galois group = S5

Hermite, Klein : resolution of the fifth degree equation by elliptic functions

2-The Klein’s Quartic X(7) and its group G168

Equation :X(7) ={[x:y:z] \in P2(C) ; x3y+y3z+z3x = 0}

A smooth algebraic curve, riemannian, of genus 3

Its group of homographic symetries is G168, the only simple group of order 168. So we get the maximal od symetries in genus 3.

QuickTime™ et undécompresseur TIFF (non compressé)

sont requis pour visionner cette image.

QuickTime™ et undécompresseur TIFF (non compressé)

sont requis pour visionner cette image.

Relation to the borromean link

Relation to the borromean field

In fact G168 = GL3(GF(2)) = GL(GF(8)/GF(2))

Equation :X(7) ={[x:y:z]  P2(C) ; x3y+y3z+z3x = 0}

Group of automorphisms = GL3(GF(2)) = G168

Generators:

€

1 0 1

1 1 1

1 1 0

R = S =

€

1 1 1

1 0 1

0 1 1

I =

€

0 1 1

1 1 0

1 1 1Relations:

(SRIR-1)2 = 1S7 = 1

((IS3I-1)(SRIR-1))3 = 1

((IS3I-1)4(SRIR-1))4 = 1

+ R

SI

R, S, I as permutations on {1, 2, 3, 4, 5, 6, 7} : R=(1746325), S = (1647235), I= (1564327).

T=(142)(356)

TRT-1 = STST-1 = ITIT-1 = R

A borromean group: G168

Equation :X(7) ={[x:y:z]  P2(C) ; x3y+y3z+z3x = 0}

Group of automorphisms = GL3(GF(2)) = G168

R, S, I as permutations on {1, 2, 3, 4, 5, 6, 7} : R=(1746325), S = (1647235), I= (1564327).

T=(142)(356) TRT-1 = STST-1 = ITIT-1 = R

A = (46)(57)B = (23)(67)C= (15)(37)

TAT-1 = BTBT-1 = CTCT-1 = A

R = ACBS = BACI = CBA

IR =: At

RS =: Bt

SI =: Ct

A = RtIt

B = StRt

C = ItSt

R

S I

Bt =RS IR = At

SI = Ct

A borromean group: G168

3. Moving Logic

Theory of Meaning

Theory of True=

Galois

Boole

3. Moving Logic (continuation)

2n as a (unique) boolean algebra, 2n as a (unique) field ;so what is the relation between these two unique structures on 2n ?

Let e = (e1, …, en) be a basis of GF(2n) over GF(2)We get on 2n a conjonction e

€

( x iei)∧e∑ ( x je j ) = ( x iy iei)∑∑and a negation e :

,

€

¬e x = x + te with

€

te = e1 + ...+ en .

€

x ⇒ e y = (¬ e x)∨e y

Could we recovered the field multiplication from this boolean structure ?

No, but…

…We have to thing to the system of all the e where e moves in the set of basis, as a moving logic,i.e. a logical analoguous of the moving frame* :

x me y = m(m-1x e m-1y )

and to use of 2n equipped with the system of all these e as a logical manifold.

*: cf. Serret-Frenet, Darboux, Ribaucour, Cartan, Ehresmann

Key starting observation: In boolean algebra we have x2 := xx = xIn a Galois field of car. 2 we have x2 x (indiscernible), x2 x ex

It can be proved (see « Moving Logic, from Boole to Galois », in Cahiers) that:

1)In GF(2n) every function f: GF(2n)m GF(2n) is a composition of constants, , (associated to a normal basis) and the Frobenius (-)2.

2) In GF(2n) there exist 4 basis p, q, r,s such that every function f: GF(2n)m GF(2n) is a composition of constants, p, p, q, q, r, r, s, s.3) In particular of course the multiplication of GF(2n) could be written in this way.

The idea of this general theorem in fact comes from computations with borromean things and Klein’quartic hereover.

3.1. Every function is a moving boolean function

Let look at the case n = 2, i.e. GF(4)

In this case we can compute that:

€

x∧e y = x 2y 2 + te (x 2y + xy 2)

If u and v are the two roots of X2+X+1, we consider the three basisk = (u,v), a = (1, v), b = (u, 1), and we have:

€

x 2 = x∧k 1+ x∧a 1+ x∧b 1

€

xy = x 2 ∧k y + x∧k y 2 + x 2 ∧k y 2

Remark : So, as x2 x (indiscernible), the product of the Galois Field xy is such that

xy xky

Case n = 3, i.e. GF(8) = GF(2)[X]/(X3+X2+1)

r, s, i roots of X3+X2+1= 0: GF(8) = {0, 1, r, s, i, r-1, s-1, i-1}

We consider the three matrices R, S, I which generates G168 ;They provide three auto-dual basis, denoted again by

We have K = (r, s, i) as the unique normal basis of GF(8)

R = (r-1, i-1, 1) S = (1, s-1, r-1) I = (s-1, 1, i-1)

We have tR = r, tS = s, tI = i,

€

x∧R y = ix 4 + s−1x 2 + i−1x

x∧S y = rx 4 + i−1x 2 + r−1x

x∧I y = sx 4 + r−1x 2 + s−1x

€

x 2 = x∧R i + x∧S r + x∧I s

€

xy = x 2 ∧K y 2 + x∧K y 4 + x 4 ∧K y + x 4 ∧K y 2 + x 2 ∧K y 4

Remark : So, as x2 x (indiscernible), the product of the Galois Field xy is such that

xy xky

3.2. Parametrization of the family of boolean structures

In GF(4):

€

x∧t y = x 2y 2 + t(x 2y + xy 2)

In GF(8):

€

x∧e y = x 4 y 4 + l−5m[x 4 y 2 + x 2y 4 ] +

€

l−4 (t +1)[x 4 y + xy 4 ] + l−2t[x 2y + xy 2]

with : q = (e1+e2+e3)(e1e2+e2e3 +e3e1)(e1e2e3)-1

t = (q+1)4, m = 1+t2+t3+t4+t5+t6+t7, l = (e1+e2+e3)t-1.

with t = e1+e2

3.3. Logical differentials or cohomological theory of meaning

- It is very good, but I dont like it »

G but Not G

Speculation of the Meaning : From the common general point of view C it is very good to eat it,andFrom my individual point of view I it is not good to eat it.

(E C G) ?(E I ??G)

Paradoxal ? (G) (G) = 0 ! Logical expression

Moving Logic expression

« - Do you like this cake ?

So, in this sentence, « but » is interpreted as:

but = (E C (-)) ?(E I ??(-))

and the meaning lives in the move of indices (C, ?, I, ??) = (ae, e+be, ce, de) :

You can compute in GF(4) with X tY = X2Y2+t(X2Y+XY2+1).

X tY = X2Y2+t(X2Y+XY2).

tX = X+t

[X t+hY - X tY]/h.

In fact the pure meaning is the Logical Difference Equation

(E e+a G) e+c(E e+b e+dG) - (E e G) e(E e eG) = 1

which can be expressed through logical derivative like the derivative of implication

Meaning(G but Not G) = {(a, b, c, d) ; (E ae G) ce (E bedeG) = 1}

There you can use of additive notations for indices

E N D