Results in Mathematics

, Volume 63, Issue 1–2, pp 597–610 | Cite as

Reflection Spaces and Corresponding Kinematic Structures

Article

Abstract

For a reflection space (P, Γ) [introduced in Karzel and Taherian (Results Math 59:213–218, 2011)] we define the notion “Reducible Subspace”, consider two subsets of \({\Gamma, \Gamma^{+} := \{a b\,|\, a,b \in P\}}\) and \({\Gamma^{-} := \{a b c\,|\, a, b, c \in P\}}\) and the map
$$ \kappa : 2^{P} \to 2^{\Gamma^+} ; X \mapsto X \cdot X := \{xy\,|\, x,y \in X\}$$
We show, for each subspace S of (P, Γ), V := κ(S) is a v-subgroup (i.e. V is a subgroup of Γ+ with if \({\xi = xy \in V, \xi \neq 1}\) then \({x \cdot \overline{x,y}\subseteq V}\)) if and only if S is reducible. Our main results are stated in the items 1–5 in the introduction.

Mathematics Subject Classification (2000)

Primary 51M10 Secondary 51G05 

Keywords

Reflection space Kinematic space Incidence group Reducible set v-Subgroup 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenMunichGermany
  2. 2.Department of Mathematical SciencesIsfahan University of TechnologyIsfahanIran

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