Über pseudoaffine Räume. (On pseudoaffine spaces)
A partial line space all of whose subplanes are affine is called pseudoaffine. A pseudoaffine space must be affine if its order exceeds three. The author treats pseudoaffine spaces \((P,{\mathcal L})\) of order 3. Such spaces are shown to correspond biuniquely to commutative Moufang loops of exponent 3. Using this method of coordinatization the author solves three problems which were posed by H. Karzel [cf. \textit{H. Karzel} and \textit{I. Pieper}, Jahresber. Dtsch. Math.-Ver. 72, 70-114 (1970; Zbl 0202.510), p. 85, and \textit{H. Karzel}, Mitt. Math. Ges. Hamb. 11, 355-368 (1985; Zbl 0593.51016), p. 355]. The results are: (1) In some pseudoaffine space \((P,{\mathcal L})\) there exists a line which is not central, although its set of parallels is left fixed by some translation (:=automorphism having no fixed points) of \((P,{\mathcal L})\). (\(L\in {\mathcal L}\) is called central iff \(X\| L\), \(Y\| L\) implies \(X\| Y\) for all \(X,Y\in {\mathcal L}\).) (2) Any pseudoaffine Euclidean space must be affine. (3) Let (H,\(\cdot)\) be a group of exponent 3. \(x,y:=\{x,y,x\bar y^ 1x\}\), \({\mathcal L}:=\{\overline{x,y}: x,y\in H\), \(x\neq y\}\). Subspaces of the pseudoaffine space \((H,{\mathcal L})\) are called z-pseudoaffine, and the author characterizes them by the following Theorem: Let \((P,{\mathcal L})\) be a partial line space. Equivalent are: (a) \((P,{\mathcal L})\) is \(z\)-pseudoaffine. (b) \((P,{\mathcal L})\) is coordinatized by some commutative Moufang loop of exponent 3 and nilpotency class at most 2. (c) For any \(A,C\in {\mathcal L}\), \(A\| C\), and \(x\in P\) such that \(\{x\| A\}\neq \{x\| C\}\) (\(\{p\| L\}\) denotes the unique line parallel to \(L\), containing \(p\)), the plane generated by \(\{x\| A\|\}\), \(\{x\| C\}\) contains (up to parallelism) exactly one central line.
scientific article; zbMATH DE number 4042684
scientific article