An extension of a theorem of A.D. Aleksandrov to a class of partially ordered fields

In [1.] established a theorem about the linearity of maps, preserving partial orders (obtained from causal relations) on space-time. In 1964 it was partly reproved by E. C. Zeeman. For one of the cases, considered by Aleksandrov, the theorem was generalized by the first-named author to arbitrary commutative fields. In the present paper, a generalization of this theorem is proved for fields with characteristic ¿ 2; a counterexample of the generalization is constructed for F2 Moreover some counterexamples of the 1974 theorem are given for Hermitean forms. The main part of the present paper consists of an extension of the other cases of Aleksandrov's theorem to a class of partially ordered fields. Finally some theorems are proved about the transitivity of the group G of causal automorphisms on some subsets of V.