Valuations on PTR's induced by triangle buildings
Recently, \textit{J. Tits} [Lect. Notes Math. 1181, 159-190 (1986; Zbl 0611.20026)] classified all affine buildings of rank \(\geq 4\), they come from algebraic groups over a local field. There are affine buildings not arising in this way, the so-called non-classical ones. Their Buekenhout diagram is of three possible types. There are examples of non-classical buildings for each of the three possible types. The present paper describes the triangle buildings (i.e. buildings whose Buekenhout diagram is a triangle) as follows: There is a bijective correspondence between the class of triangle buildings (with a complete set of apartments) and the class of projective planes coordinatized by a complete planar ternary ring with valuation.
scientific article; zbMATH DE number 4042692