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

Abstract

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 τ 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 τ induces on ${\overline{K}}$ a cyclic order ω 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 ${a\in K\,a < \infty}$. Then (K, <) is a totally ordered set. We show there is a surjective distance function:

$$ \lambda : P \times P \to \overline{K}_+ := \{x \in \overline{K}\,|\,o \leq x\leq\infty\}, $$

with “${\lambda(a,b) = \lambda(c,d) \ \Longleftrightarrow (a,b) \equiv (c,d)}$”. We prove in the first part of our project like (cf. Gröger in Mitt Math Ges Hamburg 11:441–457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with λ(a, c), λ(b, c) < e then λ(a, b) ≤ λ(a, c) + λ(b, c) < ∞.