Reducts of some structures over the reals

We consider reducts of the structure ℛ = 〈ℝ, +, ·, a 〉 a ∈ ℝ and ℛ, and we demonstrate how to recover the definition of multiplication in more general contexts than the semialgebraic one. We then conclude a similar result for reducts between 〈ℝ, ·,