The defect in an invariant reflection structure

Abstract

The defect function [introduced in Karzel and Marchi (Results Math 47:305–326, 2005)] of an invariant reflection structure (P, I) is strictly connected to the precession maps of the corresponding K-loop (P, +), therefore it permits a classification of such structures with respect to the algebraic properties of their K-loop. In the ordinary case (i.e. when the K-loop is not a group) we define, by means of products of three involutions, four different families of blocks denoted, respectively, by ${\mathcal{L}_G, \mathcal{L}, \mathcal{B}_G, \mathcal{B}}$ (cf. Sect. 4) so that we can provide the reflection structure with some appropriate incidence structure. On the other hand we consider in (P, +) two types of centralizers and recognize a strong connection between them and the aforesaid blocks: actually we prove that all the blocks of (P, I) can be represented as left cosets of suitable centralizers of the loop (P, +) (Theorem 6.1). Finally we give necessary and sufficient conditions in order that the incidence structures ${(P, \mathcal{L}_G)}$ and ${(P,\mathcal{L})}$ become linear spaces (cf. Theorem 8.6).