The Schwan/Artin coordinatization for nearfield planes (Q1336202)

Let \((M,+)\) be a group, \(\pi\) a nontrivial fibration (= partition in subgroups) and let \(X,Y \in \pi\) such that \(X \neq Y\) and \(M = X + Y\). Then the author calls the set \[ \begin{multlined} {\mathcal N}(M,\pi,X,Y) := \{\alpha : M \to M\mid \forall U \in \pi : \alpha (U) \subset U,\\ \forall(x,y) \in X \times Y : \alpha(x + y) = \alpha(x) + \alpha(y)\}\end{multlined} \] the nucleus of \((M,\pi)\) with respect to \(X\), \(Y\), and he proves: (i) \(({\mathcal N} \setminus \{0\},\circ)\) is a monoid and \(\alpha \in {\mathcal N} \setminus \{0\}\) is injective. (ii) \(({\mathcal N},+,\cdot)\) is a zero symmetric nearring, (iii) If \((M,\pi)\) is an \((X,Y)\)-spread, i.e. \(X,Y \triangleleft M\) and \(X + U = Y + V = M\) for all \(U \in \pi \setminus \{X\}\), \(V \in \pi \setminus \{Y\}\), then \(({\mathcal N},+,\cdot)\) is a nearfield; and if furthermore ``(NT) \(\exists U \in \pi\) s.t. \({\mathcal N}^* \cdot u = U^*\) for \(u \in U^* := U \setminus \{0\}\)'' is valid and \((M,\pi)\) is a spread, then \(\mathcal N\) is a planar nearfield. With these observations he obtains the following characterization theorem: For a translation plane \(\mathfrak A\) the following assertions are equivalent: (1) \(\mathfrak A\) is isomorphic to a translation plane over a planar nearfield. (2) There are two distinct infinite points \(p\), \(q\) and a finite point 0 s.t. \(\mathfrak A\) is \((p,\overline {0,q})\)-transitive. (3) There are two lines \(X\), \(Y\) s.t. \(X \nparallel Y\) and \({\mathcal N}(T,\pi,X,Y)\) of the translation spread \((T,\pi)\) satisfies axiom (NT).