Collineations of Affine Moulton Planes

Abstract

In a recent article on Moulton planes (8), I have generalized the non-Desarguesian planes introduced by F. R. Moulton (6) and Pickert (7, pp. 93-94). Let F = (0, 1; a, b, . . . , x, y, . . .} denote a given field, having P as a multiplicative subgroup of index 2. Define x > 0 or x < 0 according as x lies in P or in the other coset of non-zero elements. The function ϕ is order-preserving, and can be assumed to satisfy ϕ(0) = 0, ϕ(1) = 1. For any n < 0, the maps x —> ϕ(x) and x —> ϕ(x) — nx both carry F onto itself. The elements of F form a Cartesian group, {+,0} being defined so that