Spiegelungen in Hjelmslev'schen Kreisgeometrien. (Reflections in Hjelmslev's circle geometries)
The article aims at a synthetic approach to inversive geometry over a commutative local ring such that 2 is a unit, every non-unit is a zero- divisor, and the ideals form a chain. The coordination theorem requires the following concepts and assumptions. \(E=(P,Z,\beta,\in)\) consists of the point-set P, the set of cycles Z, a relation \(\beta\) \(\subseteq Z\times P\times Z\) (touching of cycles), and incidence \(\in\). Two points A, B are called neighbors (A\(\sim B)\) if for every point X there are O or more than 1 cycles passing through A, B, X. \(A^ 0\) denotes the set of neighboring points of A. For \(A\in P\) let \(Z_ A:=\{z\setminus A^ 0:\) \(z\in Z\) and \(A\in z\}.\) 1. For every point A a relation \(\|_ A\) is given such that \(E_ A=(P\setminus A^ 0,Z_ A,\|_ A)\) is an affine Hjelmslev-plane (admitting an epimorphism onto an ordinary affine plane and such that two points have at least one common line). 2. For A,B\(\in P\) the neighbor- relations of \(E_ A\) and \(E_ B\) coincide on the intersection. 3. For \(A\in P\) and x,y\(\in Z\) one has (x,A,y)\(\in \beta\) if and only if \(A\in x\cap y\) and \(x'\|_ Ay'\) in \(E_ A\) where \(x':=x\setminus A^ 0\) and \(y':=y\setminus A^ 0\). 4. The Fano-axiom holds for every \(E_ A\). 5. Every cycle contains at least 8 pairwise non-neighboring points. 6. Every cycle z admits an automorphism \(\sigma\) of E such that every point of z is fixed and \(A\sigma\) \(\nsim A\) for at least one point. 7. Let \(\sigma\) be an automorphism as in 6. and let \(A\sigma\) \(\nsim A\). Then \(x\sigma =x\) for every cycle x passing through A,A\(\sigma\).
inversive geometry over a Hjelmslev-ring
scientific article; zbMATH DE number 3962604