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 Combinatorial Fields

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nick
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PostSubject: Combinatorial Fields   Thu Aug 25, 2011 12:04 pm

hi, here is another good news,

I just found in the Carmichael book 1937
http://books.google.ca/books/about/Introduction_to_the_theory_of_groups_of.html?id=sd5QAAAAMAAJ

chapter XIII is dedicated to the one-to-one correspondence between sharply double transitive groups and near-fields.

hence, we have another algebraical structure to be described in combinatorial terms:

aNear-field' = aGroup

recall that
aField' = aCyc (defining the combinatorial fields)

take the tetrahedron as example : there are 12 rotations that permute 4 vertices
Tetrahedron' = Cyc[3]
after fixing a vertex with the finger, the remaining liberty of rotation (relabeling) is a cyc[3]


here (page 7) is shown how to introduce coordinates (is a very fast job)
http://books.google.com/books?id=7DdJaZcJ0bIC&printsec=frontcover&dq=near+rings&hl=en&ei=U9BXToPdK6nb0QHYyZjDDA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDQQ6AEwAQ#v=onepage&q&f=false

- choose a 0
- choose a 1

since the structure there is a stick-two-stuck-all structure (a combinatorial field), every point got now an identity, a unique determination.

Let' take another example, a triangular prism with 6 vertices and 6 rotation,
If we choose a vertex (combinatorial stuff) and we name it - let's say A, the other vertices became
- the left side of A one
- the right side of A one
- the correspondent of A in the other triangle
- the correspondent of the left side of A one
- the correspondent of the right side of A one

We have a name-one-all-named process.

my conclusion is

If someone wants to do what teacher Descartes asked, i.e. to introduce coordinates, he will need stick-one-stuck-all structures, and stick-two-stuck-all structures,
i. e. the primitives of Lin species.

After pinning a 0 (and a 1), all the other points get an identity, a unique determination.

Being well determined, one can carry now further algebraical operations.
In fact, the primitives of Lin, (stick-one-stick-all) structures are studied for 150 years now, and they are known today as groups and fields.

In the finite area, the record of sharp-transitivity are detained by the Mathieu groups
M11 - the only stick-four-stuck-all group
M12 - the only stick-five-stuck-all group
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nick
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PostSubject: Re: Combinatorial Fields   Wed Aug 31, 2011 3:03 am

Fano_plane'' = X.Klein_four-group

After pinning any two points in the Fano plane, a third one is determined.

The remaining geometry is described by the Klein group acting on itself (also, the P4bic species).


======================

here is another remark, that could be useful to explain what a combinatorial field (and, thus, a field) is :

the so called Complex Plane is not a plane, but is the complex line.

Yes, that surface on the blackboard, containing a C letter, a circle, a 0, a 1 and two axes is a LINE, not a PLANE.





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PostSubject: Re: Combinatorial Fields   Thu Sep 08, 2011 11:40 am

(modified - I have opened a new topic on projectivity)



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Last edited by nick on Tue Sep 20, 2011 5:45 am; edited 1 time in total
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PostSubject: Re: Combinatorial Fields   Thu Sep 15, 2011 9:35 pm

let me mention some other ones,

the stick-3-stuck-all structures are already known, they are the projective lines : http://en.wikipedia.org/wiki/Projective_line


but a nice surprise is

VectorSpace[n^k]' = Cyc[n-1](Lin [k]) = Cylinder

where k(n-1) + 1 = n^k,

yes, the container for vector spaces is a primitive of a cylinder,

given a vector (affine) space V over a field K, one may built the transformations

x ----> ax+b where
a is a coefficient in K
b is a vector in V

this is an imprimitive permutation group, with transitivity higher than 1 and lower than 2.

A vector (affine) space is the group #5 of degree 9 in the Conway (Maple) table; the blocks in the table are representatives of lines parallelism classes.

If we pin a point in a normal R^2 space, all the translations are killed. However, it remains the overall scaling with some lambda (warning, not rotations, a vector space with rotations is something else)

==============
here is a nice notation for a cyc(lin) :

start from a permutation - let's say (1 2 3)(4 5 6)(7 8 9)(0 A B) and optimize like this :

(1470 258A 369B) <------- this is a cyc3(lin4),

the derivative of 9T5 is something like (1234 4678).
there are four lines 14, 26, 37, 48 passing through 0, and after pinning the 0 one can synchronously permute the 8 symbols



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