When does a planar ternary ring uniquely coordinitise a projective plane?

Question

From every projective plane a coordinitisation can be constructed on a planar ternary ring, and conversely from every planar ternary ring a projective plane can be constructed. (For background see Weibel's survey of non-Desarguian planes).

Isomorphic planar ternary rings yield isomorphic projective planes, however there exist projective planes that can be coordinitised by non-isomorphic planar ternary rings.

Which planar ternary rings coordinitise their projective plane uniquely up to isomorphism?

(In other words there is a surjective function from isomorphism classes of planar ternary rings to isomorpism classes of projective planes; on what domain is it injective?)

As stated in Weibel's survey above two ternary rings are isomorphic if (and only if) the automorphism group of the projective plane maps any quadrilateral into any other quadrilateral. It is not clear to me, however, what this means about the ternary ring itself.

I know this class includes the alternative division rings (see Bruck and Kleinfeld - The Structure of Alternative Division Rings, Theorem B in Section 5).

P.S. I originally stated that this class contains the near-fields, but this is false.

Answer 1

The answer to the problem is well known; since :

all the PTRs coordinatizing a given projective plane are isomorphic iff it is a moufang plane iff all these PTRS are isomorphic alternative division rings. http://www.math.uni-kiel.de/geometrie/klein/math/geometry/moufang.html

However the problem to give a purely algebraic proof of this result seem to be an open problem.

Answer 2

Too long for a comment.

Contrary to what wikipedia and the accepted answer say, the following paper proves that there are non-Moufang infinite projective planes where the collineation group is transitive on quadrangles:

• Otto H. Kegel, Adolf Schleiermacher, Amalgams and embeddings of projective planes, Geometriae Dedicata 2 (1973), pp 379-395 http://dx.doi.org/10.1007/BF00181482

See also the last theorem in

• Martin Funk, Karl Strambach, On free constructions, manuscripta mathematica 72 (1991), pp 335-374, https://doi.org/10.1007/BF02568284, ResearchGate pdf

On the other hand, the collineation group is sharply transitive on quadrangles iff the plane is pappian:

• T. Grundhfer, Projective planes with collineation groups sharply transitive on quadrangles, Archiv der Mathematik, 1984 https://doi.org/10.1007/BF01190963

Answer 3

A necessary and sufficient condition so that two planar ternary rings coordinatize isomorphic projective planes was obtained in 2004 by A.Grari

• A necessary and sufficient condition so that two planar ternary rings induce isomorphic projective planes. Arch. Math. 83, (2004) 183–192, doi:10.1007/s00013-003-4580-9.

The well-known theorems on skew-fields and division rings and alternative division ring can be deduced from this condition as particular cases.

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Response added later:

Let $\pi$ be a projective plane coordinatized by a unique (up to isomorphism) Hall ternary ring $S$. If $\pi$ is finite then by the theorem of Ostrom-Wagner (see eg Kantor's 1985 Flag-Transitive Planes) $\pi$ is desarguesian; hence $S$ is a skew-field, and therefore there is no problem to solve in this case since the solution is well known.

If $\pi$ is infinite then $\pi$ is a flag-transitive plane so all the PTRs coordinatizing this plane are equivalent under the operation of comparison (see lemma 4.1 of Grari's paper above).

The problem posed can now be reduced to the following: When are two equivalent (under comparison) Hall ternary rings isomorphic?