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 Γ, Γ+ := {ab | a, b ∈ P} and
Γ− :={abc | a, b, c ∈ P} and the map κ : 2^P → 2^Γ+; X -> X · X := {xy | x, y ∈ 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 ξ = xy ∈ V, ξ not= 1 then x · x, y ⊆ V ) if and
only if S is reducible.