The undefinability of intersection from perpendicularity in the three- dimensional Euclidean geometry of lines (Q1801633)

\textit{W. Schwabhäuser} and \textit{L. W. Szczerba} [Fundamenta Math. 82(1974), 347-355 (1975; Zbl 0296.50001)] have shown that Euclidean 3- dimensional geometry is axiomatizable by using the two binary predicates (among variables to be interpreted as `lines') of perpendicularity (i.e. two lines intersecting at a right angle) and intersection. The question whether perpendicularity could serve as the only primitive notion (as is the case for dimensions \(\geq 4)\), which was left open, is answered negatively in the paper under review. This proves that both perpendicularity and intersection are needed in order to axiomatize 3- dimensional Euclidean geometry. \{Reviewer's remark: An algebraic characterization of perpendicularity- preserving permutations of Euclidean spaces was provided by \textit{W. Benz} and \textit{E. M. Schröder} [Geom. Dedicata 21, 265-276 (1986; Zbl 0605.51003)]; this result was nevertheless not available to the author when the paper was written (the reviewer saw a copy of it in 1986)\}.