Construction: Suppose that P is projective plane that satisfies
Desaragues axiom. Let L be an arbitrary line of P and p be an
arbitrary point on L. Define D = L - {p}.
Question: Proof that possible to define on D two operations (D,+,?),
so D will be division ring and each element is reversible.

What does "reversible" means ?

In the context (division ring), I feel sure cyberdima-ga means "invertible".

regards, mathtalk-ga

A basic introduction to the construction of "ternary rings" from
projective planes is found in this column:

[This Week's Finds in Mathematical Physics (Feb. 9, 2000) -- John Baez]
http://math.ucr.edu/home/baez/week145.html

A quick summary:  Projective planes are incidence structures, where
points and lines play dual roles.  E.g. two points determine a line
incident with both points, and two lines determine a point where both
lines intersect.

The column linked above also defines a Desarguesian projective plane
and describes (without proof) the equivalence of this geometric
property to the "ternary ring" constructed from the projective plane
being a division ring.

A proof of Desargues' Theorem for the real projective plane, using
vector algebra, is given in the Wikipedia article:

[Desargues' theorem -- Wikipedia]
http://en.wikipedia.org/wiki/Desargues%27_theorem

which also provides a link to an alternative proof (also using vector
algebra) at PlanetMath:

[Proof of Desargues' theorem  -- PlanetMath]
http://planetmath.org/?op=getobj&from=objects&id=4514

A proof that a projective plane constructed from a division ring is
Desarguesian can be similar to the above calculations in use of vector
algebra.  Proof of the converse is more challenging.  There is a proof
given (in PostScript format) here in Section 6.4 based on some
"machinery" of perspectivities in projective planes and a concept of a
Desarguesian projective lines:

[6. Projective and affine spaces]
www.win.tue.nl/~amc/buek/B06.ps


regards, mathtalk-ga