A Note on 3×3-valued Łukasiewicz Algebras with Negation - Gallardo, Carlos; Ziliani, Alicia (Bulletin of the Section of Logic, 2021)
In 2004, C. Sanza, with the purpose of legitimizing the study of \(n× m\)-valued Łukasiewicz algebras with negation (or \(NS_{n× m}\)-algebras) introduced \(3 × 3\)-valued Łukasiewicz algebras with negation. Despite the various results obtained about \(NS_{n× m}\)-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated \(NS_{3 × 3}\)-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \(Λ(NS_{3× 3})\) of all subvarieties of \(NS_{3× 3}\) and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \(NS_{3× 3}\).