Quasi-ordered affine planes and ternary rings (Q1180141)

In J. Geom. 41, No. 1/2, 193-202 (1991; Zbl 0733.51010) the author has introduced the notion of quasi-orderings in desarguesian spaces as a common generalization of the concepts of semiorderings and Sperner's half-orderings, showing that its algebraic counterpart, the quasi- orderings of (skew) fields \((F,+,\cdot)\), is given by the mappings \(\sigma: F\backslash\{0\}\to\{+1,-1\}\) fulfilling \(\sigma(f)\cdot\sigma(- f)=\sigma(1)\cdot\sigma(-1)\) for all \(f\in F\backslash\{0\}\). In the paper under review, the author transfers her concepts to arbitrary affine planes \(\mathcal A\). Due to the inhomogeneity of the underlying planar ternary rings (ternary fields) \((F,T)\), the algebraic counterparts of quasi-orderings of \(\mathcal A\), the quasi-orderings \(\sigma: F\backslash\{0\}\to\{+1,-1\}\) of \((F,T)\), additionally have to satisfy: \[ \sigma(T(a,b,c)-T(a,b,d))=\sigma(c-d)\qquad\hbox{for all}\qquad a,b,c,d\in F,c\neq d, \] \[ \sigma(T(a,b,d)-T(a,c,d)=\sigma(ab- ac)\qquad\hbox{for all}\qquad a,b,c,d\in F,a\neq 0,b\neq c. \] However, the author's notion is weak enough to guarantee that any Cartesian group \(\neq GF(2)\) with an abelian addition admits proper non-trivial quasi- orderings, and that in each of the Lenz-Barlotti classes which are known to be not empty there exist affine planes carrying proper non-trivial quasi-orderings.