Written exam MPRI 2 - 6 , year 2014 – 2015

αU (R) ⊆ R] ⇐⇒ { (ρ, a(i)) | (ρ, a) ∈ R, i ∈ [0, ρ(N)− 1] } ⊆ R] ⇐⇒ ∀(ρ, a) ∈ R : ∀i ∈ [0, ρ(N)− 1] : (ρ, a(i)) ∈ R] ⇐⇒ (ρ, a) ∈ R =⇒ ∀i ∈ [0, ρ(N)− 1] : (ρ, a(i)) ∈ R] ⇐⇒ R ⊆ { (ρ, a) | ∀i ∈ [0, ρ(N)− 1] : (ρ, a(i)) ∈ R] } ⇐⇒ R ⊆ γU (R]) Actually, we have a Galois embedding as αU is onto: any (ρ, x) ∈ EU is the abstraction αU ({ (ρ, a) }) of some (ρ, a) ∈ E such that ∀i ∈ [0, ρ(N)− 1] : a(i) = x.