Reflection Spaces and Corresponding Kinematic Structures

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.