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 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. `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``` 