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A note on Pappus’ theorem. (English) Zbl 1027.51002
Let \(\Pi\) be a projective plane coordinatized by a ternary ring \((R,T)\). For three lines \(u,v,w\) in \(\Pi\) the \((u,v,w)\)-Pappus’ theorem holds if the usual Pappus configuration on the lines \(u,v\) with points of intersection on \(w\) closes for all suitable sets of points.
The authors prove two theorems. (1) In \(\Pi\) the \(([0],[\infty],[1])\)-Pappus theorem holds iff the operation \(a\ast b=T(a,1,b)\) is commutative and associative, i.e., \((R,\ast)\) is an abelian group.
(2) Assume that \(a\ast b=T(1,a,b)\) then the \(((0),(1),(\infty))\)-dual-Pappus’ theorem holds iff \(\ast\) is commutative and associative.
MSC:
51A35 Non-Desarguesian affine and projective planes
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