Semi-classical projective planes over half-ordered fields (Q1900106)

In 1992 \textit{J. Jakóbowski} [ibid. 42, No. 3, 243-253 (1992; Zbl 0756.51007)] has given a vast generalization of Pierce's (and Pickert's) construction of Moulton planes over (half-)ordered fields, which in the main works by bending the lines of the half-ordered desarguesian plane over \(F\) in only one of the affine halfspaces of the \(y\)-axis. The author shows that, up to dualization, these semi-classical planes over half-ordered fields \(F\) are coordinatized by ternary fields \((F,T)\) the ternary operation of which is given by \[ T(a,x,b) := \begin{cases} ax + b & \text{if \(x \geq 0\)}\\ g^{-1} (h(a) g(x) + g(b)) & \text{if \(x < 0\),} \end{cases} \] where \(g\) and \(h\) are halforder-preserving permutations of \(F\) fixing 0 and 1 and additionally fulfilling that for all \(a, b, c \in F\) with \(c < 0 < a\) the function \(x \mapsto g(ax + b) + ch(-x)\) from \(F\) to itself is surjective. This characterization of semi-classical projective planes allows the author to explicitly determine all isomorphisms between such planes which map the lines at infinity and the \(y\)-axes onto each other (possibly switching them). Each of these isomorphisms turn out to be a concentration of an isomorphism induced by a linear map, an isomorphism induced by an isomorphism between the underlying halfordered fields, and possibly an isomorphism that interchanges the roles of the two halfspaces and/or switches the two lines mentioned above. For the special case of ordered, proper semi-classical projective planes, the author is able to prove, that there exist no other isomorphisms; a fact, which finally leads to the complete solution of the isomorphy problem for these planes: Two ordered semi-classical projective planes, given by \(g\), \(h\) over \(F\) and by \(g'\), \(h'\) over \(F'\) respectively, are isomorphic as ordered projective planes, if and only if \((h', g')\) is positively affinely equivalent to \((g,h)\) or to \((h,g)\) or negatively affinely equivalent to \((h^{-1}, g^{-1})\) or to \((g^{-1}, h^{-1})\). Herein two pairs of permutations \((g,h)\) of \(F\) and \((g', h')\) of \(F'\) are called positively (or negatively) affinely equivalent, if there exist order-preserving isomorphisms \(\Phi\), \(\Psi\) from \(F\) onto \(F'\) and elements \(a, b, c, d, a^*, b^*, c^*, d^* \in F\) with \(c, c^* \neq 0\) and \(aa^* > 0\) (resp. \(aa^* < 0)\) such that \(h'(\Phi(x)) = \Psi(\text{ch} (ax + b) + d)\) and \(g'(\Phi(x)) = \Psi(c^*g (a^* x + b^*) + d^*)\) for all \(x \in F\).