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I made projective plane less stubby.
A colleague of mine worked on a synthetic theory of projective planes over local rings for his thesis. If I recall correctly, he needed to add a variant of Pappus’s theorem as an axiom.
Is there anything available about it?
Wasn’t that presented at CT2014?
There’s also the result of Ruth Moufang about coordinatising projective planes satisfying (edit: little) Desargues with alternative division rings. She discovered $OP^2$ this way, I gather.
One thing I don’t yet understand: a projective plane without any Desargues’ theorem can be constructed/coordinatized using a “ternary ring” (which isn’t actually a ring at all, but an algebraic gadget whose basic operation is ternary, thought of as like $(a,b,c) \mapsto a b + c$). Is the construction of $O P^2$ a special case of that?
$OP^2$ is a Moufang plane, so little Desargues’ holds (Theorem 3.8 here). This is strictly weaker than the usual Desargues’ theorem. Thanks for pointing out planar ternary rings. My officemate as a PhD student was a finite geometer, so I learned a bit by osmosis about non-Desarguesian planes, but not about these.
Did John Baez’s account cover this? Here he is on Octonionic Projective Geometry.
JB says
in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions
Here’s Moufang’s paper he cites:
Ruth Moufang, Alternativkörper und der Satz vom vollständigen Vierseit, Abh. Math. Sem. Hamburg 9 (1933), 207-222. doi:10.1007/BF02940648
Right, I know all that. Since $O P^2$ is non-Desarguesian, we can construct from it a ternary ring that coordinatizes it — but my question is, can the original construction of $O P^2$ from $O$ be obtained by way of regarding $O$ as a ternary ring? I presume so, but in that case it surprises me that hardly anyone seems to mention this when discussing $O P^2$, so I end up with doubt.
From the wikipedia article on Moufang planes I linked to, we find characterisation of them:
Some ternary ring of the plane is an alternative division ring.
P is isomorphic to the projective plane over an alternative division ring.
Also, in a Moufang plane:
Any two ternary rings of the plane are isomorphic.
Since O, as a ternary ring, is an alternating ring, any other ternary ring coordinatising $OP^2$ is isomorphic to O.
When you say the original construction, I guess one would have to dig up either a translation of Moufang’s construction, or read it in German. My guess is that the ternary ring version reproduces Moufang’s result in the alternating ring case in general, not just for O.
Oh good.
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