TY - JOUR

T1 - The defect in an invariant reflection structure

AU - Karzel, Helmut

AU - Pianta, Silvia

AU - Marchi, Mario

PY - 2010

Y1 - 2010

N2 - 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)

AB - 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)

KW - K-loop

KW - loop

KW - loop-derivation

KW - reflection structure

KW - K-loop

KW - loop

KW - loop-derivation

KW - reflection structure

UR - http://hdl.handle.net/10807/55298

U2 - 10.1007/s00022-010-0058-7

DO - 10.1007/s00022-010-0058-7

M3 - Article

SN - 0047-2468

VL - 99

SP - 67

EP - 87

JO - Journal of Geometry

JF - Journal of Geometry

ER -