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Abstract
In 2004, C. Sanza, with the purpose of legitimizing the study of \(n\times m\)-valued Łukasiewicz algebras with negation (or \(\mathbf{NS}_{n\times m}\)-algebras) introduced \(3 \times 3\)-valued Łukasiewicz algebras with negation. Despite the various results obtained about \(\mathbf{NS}_{n\times m}\)-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 \(NS_{3 \times 3}\)-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \(\Lambda(\mathbf{NS}_{3\times 3})\) of all subvarieties of \(\mathbf{NS}_{3\times 3}\) and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \(\mathbf{NS}_{3\times 3}\).
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