Symmetric Minkowski planes ordered by separation
In [\textit{H. Karzel, H. Kosiorek} and \textit{J. Matras}, J. Geom. 93, 116--127 (2009; Zbl 1206.51003)], an order concept for a symmetric Minkowski plane was introduced where only the set of circles is involved, namely by a valuation which associates to each circle a value which is 1 or \(-1\). Then since the set of circles is endowed with a multiplication and becomes a group, it turns out that the valuation is a homomorphism between this group and the cyclic group of order 2. Moreover, when the Minkowski plane is symmetric, the group of circles is isomorphic to the projective linear group PGL\((2,F)\), where \(F\) denotes the commutative field corresponding to the symmetric Minkowski plane. In [loc. cit.], the authors proved that there is a one-to-one correspondence between halforders of the field \(F\) and orthogonal valuation of the symmetric Minkowski plane. In this paper, the authors consider an order concept which is based on the notion of separation and order separation for quadruples of non concyclic points of the Minkowski plane. They prove that one can derive from an orthogonal valuation of a symmetric Minkowski plane a separation of the plane and this separation is harmonic iff the valuation is. Finally they prove some properties of a separation of a symmetric Minkowski plane and prove that a halfordering of the associated field can be derived from separation. Combining these results, starting from a symmetric Minkowski plane with an orthogonal valuation, the authors derive first a halfordering of the corresponding field and secondly a separation of the Minkowski plane. This separation induces a halfordering on the field which is equal to the previous one.
scientific article; zbMATH DE number 5844608