Klassifizierung von Halbordnungen. (On classifications of halforderings) (Q1107823)

Basic properties of the halforderings of affine and projective planes in the sense of \textit{E. Sperner} [Math. Ann. 121, 107-130 (1949; Zbl 0032.178)] are carried over to arbitrary linear spaces (Inzidenzräume). In particular, the classical results on Sperner's functional Z, given by \textit{H. Karzel} [Math. Z. 62, 268-291 (1955; Zbl 0066.388), Algebr. Topol. Foundations Geom., Proc. Colloq. Utrecht, August 1959, 71-86 (1962; Zbl 0107.374)] for projective planes, extend to the general case: Z takes only odd or only even values, the cases \(Z=2\) and \(Z=3\) can only occure in the classical affine planes of order 5, 3 respectively, and \(Z=1\) implies the infinity of the point set of a line.