The defect in an invariant reflection structure

Helmut Karzel, Silvia Pianta, Mario Marchi

Risultato della ricerca: Contributo in rivistaArticolo in rivistapeer review

1 Citazioni (Scopus)


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 LG,L, BG, 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,LG) and (P,L) become linear spaces (cf. Theorem 8.6)
Lingua originaleEnglish
pagine (da-a)67-87
Numero di pagine21
RivistaJournal of Geometry
Stato di pubblicazionePubblicato - 2010


  • K-loop
  • loop
  • loop-derivation
  • reflection structure


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