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⎩ x if i = 1, ¬x if i = 0, is an injective homomorphism from S(A , ¬¬) into A. It is surjective (i.e., an isomorphism onto A) iff A is directly indecomposable. As a matter of fact, ψA is one-to-one and preserves the operations , and ⇒, but it fails to preserve . Indeed, for a, b ∈ A, ψA(b, 1) ∗ ψA(¬¬a, 0) = b ∗ ¬a, ψA(〈b, 1〉 〈¬¬a, 0〉) = ψA(b → ¬¬a, 0) = ¬(b → ¬¬a). Hence it is easy to corroborate that ψA preserves , and hence it is an embedding, if and only if the following condition holds: For any a, b ∈ A , ¬(b → ¬¬a) = b ∗ ¬a. (1) Observe that ¬(b → ¬¬a) = ¬¬(b ∗ ¬a). Since ∇(b ∗ ¬a) = ⊥, (1) will hold if we require that ∇(x) = ⊥ implies ¬¬x = x.

Erratum to: Free Algebras in Varieties of Glivenko MTL-Algebras Satisfying the Equation $${2(x^2) = (2x)^2}$$ 2 ( x 2 ) = ( 2 x ) 2