Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality

This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to introduce new concepts and develop their theories. If ${(P,mathfrak{G}, equiv,tau)}$ denotes an elliptic geometry ordered via a separation ?? then there are polar points o and ?? and on the line ${ overline{K} := overline{infty,o}}$ there can be established an operation ??+?? such that ${(overline{K},+)}$ becomes a commutative group and the map ${ a^+ :overline{K}to overline{K} ; x mapsto a + x}$ is a motion on ${overline{K}}$ . The separation ?? induces on ${overline{K}}$ a cyclic order ?? with [o, e, ??] = 1 such that ${(overline{K},+,omega)}$ becomes a cyclic ordered group. For ${a,b in K := overline{K} {setminus}{infty}}$ we set ${a < b :Longleftrightarrow [a,b,infty] =1}$ and for all ${ain K,a < infty}$ . Then (K,?<) is a totally ordered set. We show there is a surjective distance function: $$ lambda : P times P to overline{K}_+ := {x in overline{K},|,o leq xleqinfty}, $$ with ?? ${lambda(a,b) = lambda(c,d) Longleftrightarrow (a,b) equiv (c,d)}$ ??. We prove in the first part of our project like (cf. Gr?ger in Mitt Math Ges Hamburg 11:441?C457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with ??(a, c), ??(b, c) < e then ??(a, b) ?? ??(a, c) + ??(b, c) < ??.