Ordered symmetric Minkowski planes I (Q1039857)

Let \({\mathcal{M}}\) be a Minkowski plane. It is well known [see \textit{R. Artzy}, J. Geometry 3, 93--102 (1973; Zbl 0245.50031), \textit{W. Benz}, Vorlesungen über Geometrie der Algebren. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0258.50024), \textit{H. Karzel}, Math. Z. 62, 268--291 (1955; Zbl 0066.38803)], that if the rectangle axiom holds then an orthogonality between circles can be defined and there exists a commutative field \(F\) such that the circles of \({\mathcal{M}}\) can be identified with the elements of the projective linear group \(PGL(2, F)\). In particular two circles \(A, B\) are orthogonal (\(A {\perp}B\)) iff \(A \neq B\) and \(A ^{ - 1} B = B ^{ - 1} A\). In the paper the case \(char F \neq 2\) is considered and the authors introduce a ``valuation'' of the circles, namely a function \(h\) that associates to each circle one of the values \(1\) or \(-1\). The main condition for the valuation \(h\) demands that for three circles \(A,B,C\), where \(A\) is orthogonal to \(B\) and \(C\) and the circle \(D= B \cdot \,A^{-1} \cdot C\) the product \(h(A)\cdot h(B)\cdot h(C) \cdot h(D)\) equals \(1\). If this condition is valid the pair \(({\mathcal{M}},h)\) is called a halfordered symmetric Minkowski plane and \(h\) an orthogonal valuation. In the paper the authors prove that there is a one to one correspondence between halforders of the field \(F\) and orthogonal valuation of symmetric Minkowski planes. In a forthcoming paper the notion of ``separation'' for quadruples of concyclic points will be derived from a valuation and so the connection with the concept of an ordered symmetric Minkowski plane given in [\textit{H. J. Kroll}, Abh. Math. Semin. Univ. Hamb. 46, 217--255 (1977; Zbl 0374.50006)] will be established.