Some chain geometries determined by transformation groups

Regular direct products of affine partial linear spaces

“…On the other hand, the structure W, C is exactly the chain geometry determined by the group (Dil(C 0 )) I acting on V I (starting from the transformation group (Dil(C 0 ), V )), as defined in [5].…”

“…(5) and (6)) the relation˘ is the transitive closure of the relation . (3), and then h(Y ) = h τ h −1 h(X) and, clearly, h τ h −1 = hτ h −1 and hτ h −1 ∈ (Tr(C 0 )) I (convention: coordinatewise composition).…”