Affine Geometry of Spine Spaces

Abstract

Abstract. The parallelity relation and the group of dilatations in the geometry of spine spaces are investigated. Fundamental theorems of affine geometry are proved and the analytical representation of dilatations is given. IntroductionThe paper is a continuation of the theory of spine spaces originated in [2] and developed in [4] and [3]. It seems that there are two approaches to the geometry of spine spaces, the projective one, with no parallelity relation involved, and the affine, where the parallelity is defined. While [4] deals with projective aspects of spine spaces, and [3] deals with affine aspects of a narrow class of spine spaces of linear complements only, this paper gives an account for general properties of spine spaces common to affine geometry.Most of results and constructions provided for spine spaces do not make use of the natural parallelity of spine spaces. The geometry of spine spaces with parallelity defined, however, resembles the affine geometry in many aspects. In order to utilize the parallelity it is necessary to make distinction between affine and projective lines. The parallelity is an equivalence relation in the set of affine lines, and two parallel lines which intersect each other coincide. It is only partial, not Euclidean, i.e. directions do not cover the point-set, but affine variants of the Veblen condition hold (3.2), as well as the stronger parallel triangle completion condition (3.3). It also turns out that the geometry of a spine space equipped with the natural parallelity satisfies fundamental theorems of affine geometry, that is Desargues theorem (3.4) holds true, and Pappus axiom holds iff the ground field is commutative (3.5).

Extended parallelity in spine spaces and its geometry

“…the extended parallelity. As a direct consequence of 6.4, 6.3, 6.2, and the results of [12] we obtain…”

“…iff f = id. Comparing the conditions of 6.3 with the characterization of dilatations presented in [12] we obtain…”

“…According to 2.7 h 2 ⊆ , and hence h 1 h 2 by 2.6. Now, the Veblen condition for , proved in [12], can be applied. Observe that y, z are linearly independent, y, z ∈ H ∩ G and y, z ∈ X.…”

“…A characterization of the dilatations of the natural parallelity of a spine space M was given in [12]. In this paper we have established an analytical characterization of the dilatation group of the relation in a spine space (cf.…”

“…Since X i carries the structure of a spine space (cf. [12]), the -horizon of X i is, up to an isomorphism, X ∞ i / τ by 4.1. According to [12] the map γ : Comp τ (U ) (Trc U, Ctr U) is a one-to-one mapping between H ∞ (A)/ τ and the product […”