A Note on 3×3-valued Łukasiewicz Algebras with Negation

Carlos Gallardo; Alicia Ziliani

doi:10.18778/0138-0680.2021.10

Authors

Carlos Gallardo Universidad Nacional del Sur, Departamento de Matemática
Alicia Ziliani Universidad Nacional del Sur, Departamento de Matemática https://orcid.org/0000-0002-2908-5218

DOI:

https://doi.org/10.18778/0138-0680.2021.10

Keywords:

n

-valued Łukasiewicz-Moisil algebras,

n \times m

-valued Łukasiewicz algebras with negation, free algebras, lattice of subvarieties

Abstract

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

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