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Q: Projective Geometry Problem ( No Answer,   3 Comments )
Subject: Projective Geometry Problem
Category: Science > Math
Asked by: cyberdima-ga
List Price: $50.00
Posted: 29 Nov 2006 08:09 PST
Expires: 29 Dec 2006 08:09 PST
Question ID: 786570
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.
There is no answer at this time.

Subject: Re: Projective Geometry Problem
From: d10s-ga on 29 Nov 2006 19:32 PST
What does "reversible" means ?
Subject: Re: Projective Geometry Problem
From: mathtalk-ga on 07 Dec 2006 15:05 PST
In the context (division ring), I feel sure cyberdima-ga means "invertible".

regards, mathtalk-ga
Subject: Re: Projective Geometry Problem
From: mathtalk-ga on 29 Dec 2006 19:54 PST
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]

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]

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

[Proof of Desargues' theorem  -- PlanetMath]

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]

regards, mathtalk-ga

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