In these particular cases, not only the notions of equivalence but also the characteristic semantic structures coincide, i.e., relativized hyperequivalence interpretations coincide with the respective characteristic sets of equivalence interpretations.

Proposition 6 Let Γ be a propositional theory over **L ⊆ A**, and let e ∈ {**a, s, u**}, A+(a) = A−(a) = ∅,
A+(s) = A−(s) = A, A+(u) = A, and A−(u) = ∅. Then, Ee(Γ) = E
A
+(e)

$$E_{A^{-}(e)}^{\alpha^{-}(e)}(\Gamma)$$

## Proof

First consider answer-set equivalence, i.e., e = a and A+ = A− = ∅. Then for any HTinterpretation (**X, Y** ), it holds that (**X, Y** ) ∈ EA
+
A− (Γ) = E
A
+(e)
A−(e)
(Γ) iff (∅, Y ) is thereclosed in Es(Γ) and X = ∅. The former follows from the first condition in Definition 9 since Y |A+ = ∅, and the latter from the second condition in Definition 9, i.e., from the existence of an X′such that X = X′|A+ (since X′|∅ = ∅ for any X′). Note that X = ∅
and (∅, Y ) there-closed in Es(Γ) are exactly the requirements for (**X, Y** ) ∈ Ea(Γ). This proves (**X, Y** ) ∈ EA
+
A− (Γ) iff (**X, Y** ) ∈ Ea(Γ).

Turning to strong equivalence, let e = s and A+ = A− = A. Then for any HTinterpretation (**X, Y** ) over A, it holds that (**X, Y** ) ∈ EA
+
A− (Γ) = E
A
+(e)
A−(e)
(Γ) iff (**Y, Y** ) in Es(Γ) and (**X, Y** ) in Es(Γ). The former follows from the first condition in Definition 9 since Y |A = Y , and the latter from the second condition in Definition 9, i.e., from the existence of an X′such that X = X′|A (which implies X′ = X since X′|A = X′for any X′) and such that X′′ ∈ Es(Γ) for all X′′ ⊆ Y where X′|A = X′′|A (i.e., for X′′ = X′ = X). Note that (**X, Y** ) ∈ Es(Γ) implies (**Y, Y** ) ∈ Es(Γ). Consequently, it holds that (**X, Y** ) ∈ EA
+
A− (Γ) iff (**X, Y** ) ∈ Es(Γ).

Eventually consider uniform equivalence, i.e., e = u, A+ = A, and A− = ∅. In this case, (**X, Y** ) ∈ EA
+
A− (Γ) = E
A
+(e)
A−(e)
(Γ), for any HT-interpretation (**X, Y** ) over A, iff
(**Y, Y** ) in Es(Γ) and (X′, Y ) in Es(Γ) for all X ⊆ X′′ ⊆ Y . The former follows from the first condition in Definition 9 since Y |A = Y , and the latter from the second condition in Definition 9, i.e., from the existence of an X′such that X = X′|A (which implies X′ = X since X′|A = X′for any X′) and such that X′′ ∈ Es(Γ) for all X′′ ⊆ Y where X′|A ⊆ X′′|A (i.e., for X′ = X ⊆ X′′ ⊆ Y ). Note that this are exactly the requirements for(**X, Y** ) being closed in Es(Γ), thus for (**X, Y** ) ∈ Eu(Γ). Therefore,(**X, Y** ) ∈ EA
+
A− (Γ)
iff (**X, Y** ) ∈ Eu(Γ), which proves the claim.

Moreover, a setting where A+ = A− is termed relativized strong equivalence, and A− =
∅ denotes relativized uniform equivalence. A further remark is in place, however. While we proved for uniform equivalence of propositional theories, that it is indifferent to whether we restrict additions (contexts) to sets of atoms or whether we allow for factual theories, A-
∅-theories syntactically do not encompass factual theories, since negation, i.e., formulas of the form a → ⊥, are not permitted. One question that this raises is: would allowing factual theories as contexts make a difference for relativized notions of uniform equivalence?

The answer is by inspection of the proof of Theorem 4 in connection with Proposition 4 and Proposition 5. Recall that the propositions have been stated for extended A+-
A−-theories. Therefore, the only-if direction of Theorem 4 also holds for extended A+-