Coordinate theorems for affine Hjelmslev planes

Abstract

It is shown that an affine Hjelmslev plane ℋ is a translation plane if and only if each of its coordinate biternary rings B=〈k, T, T0, 0, 1〉 are linear. Addition and multiplication in the ternary ring 〈k, T, 0, 1〉 are defined by a+b=T(a, 1, b) and a·b= =T(a, b, 0), respectively, and it is proved that every biternary ring of a translation plane has the additional properties that 〈k,+〉 is an abelian group 〈k, +, ·〉 is right distributive, and T(a, 1, b)=T0(a, 1, b). Moreover, if a single linear biternary ring of ℋ has these three properties, then ℋ is a translation plane. It is shown that a translation plane is Desarguesian if and only if it has a linear biternary ring such that T=T0 and 〈k, +, ·〉 is an affine Hjelmslev ring. Hessenberg’s theorem for affine Hjelmslev planes is proved, and a special configurational condition which is equivalent to the commutativity of multiplication in each biternary ring is introduced.