A subvariety of Ockham-Nelson algebras

In 1958, Rasiowa [10] introduced N-lattices (or Nelson algebras) as the algebraic counterpart of the constructive logic with strong negation ([11], [6]). Later, in 1967 Brignole and Monteiro [3], proved that N-lattices may be described as algebras A,∨,∧,→,∼, 1X of type (2, 2, 2, 1, 0) satisfying these identities: (N1) x ∧ 1 = x, (N2) x ∧ (x ∨ y) = y, (N3) x ∧ (y ∨ z) = (z ∧ x) ∨ (y ∧ x), (N4) ∼ ∼ x = x, (N5) ∼ (x ∧ y) = ∼ x ∨ ∼ y, (N6) x → x = 1, (N7) (x → y) ∧ ( ∼ x ∨ y) = ∼ x ∨ y, (N8) x ∧ (x→y) = x ∧ ( ∼ x ∨ y), (N9) x→(y ∧ z) = (x→y) ∧ (x→z), (N10) (x ∧ y)→z = x→(y→z), and the additional condition (N11) x ∧ ∼ x ≤ y ∨ ∼ y, known as Kleene law. In 1986, Cignoli [2] proved very important properties about N-algebras, which have motivated this paper. On the other hand in 1990, Figallo [4] defined generalized N-lattices, which here we call De Morgan-Nelson algebras (or DN-algebras), because in 2005 Vakarelov (see [13]) introduced under the name of generalized N-lattices a different notion from the one considered above. DN-algebras are described as algebras of the same type as N-algebras satisfying only identities (N1) t