Geodesic
A Note on 3×3-valued Łukasiewicz Algebras with Negation
Bulletin of the Section of Logic, Tome 50 (2021) no. 3, pp. 289-298.

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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.  
Keywords: \(n\)-valued Łukasiewicz-Moisil algebras, \(n \times m\)-valued Łukasiewicz algebras with negation, free algebras, lattice of subvarieties 
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Gallardo, Carlos; Ziliani, Alicia. A Note on 3×3-valued Łukasiewicz Algebras with Negation. Bulletin of the Section of Logic, Tome 50 (2021) no. 3, pp. 289-298. http://geodesic.mathdoc.fr/item/BSL_2021_50_3_a0/

[1] [1] N. Belnap, How a computer should think, Oriel Press, Boston (1977), pp. 30–56.

[2] [2] V. Boicescu, A. Filipoiu, G. Georgescu, S.Rudeanu, Łukasiewicz-Moisil Algebras, vol. 49 of Annals of Discrete Mathematics, North-Holland, Amsterdam (1991).

[3] [3] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin (1981).

[4] [4] R. Cignoli, Moisil Algebras, vol. 27 of Notas de Lógica Matemática, Universidad Nacional del Sur, Argentina (1970).

[5] [5] A. Day, Splitting algebras and a weak notion of projectivity, Algebra Universalis, vol. 5 (1975), pp. 153–162 | DOI

[6] [6] A. Day, On the lattice of subvarietes, Houston Journal of Mathematics, vol. 5 (1979), pp. 183–192.

[7] [7] A. V. Figallo, C. Sanza, The NSn×m–propositional calculus, Bulletin of the Section of Logic, vol. 37 (2008), pp. 67–79.

[8] [8] C. Gallardo, C. Sanza, A. Ziliani, F–multipliers and the localization of LMn×m–algebras, Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 21 (2013), pp. 285–304 | DOI

[9] [9] B. Jonnson, Algebras whose congruence lattices are distributive, Mathematica Scandinavica, vol. 21 (1967), pp. 110–121 | DOI

[10] [10] G. Moisil, Notes sur les logiques non-chrysippiennes, Annales Scientifiques de l’Université de Jassy, vol. 27 (1941), pp. 86–98.

[11] [11] G. Moisil, Le algebre di Łukasiewicz, Analele Universitii Bucureti, seria Acta logica, vol. 6 (1963), pp. 97–135.

[12] [12] C. Sanza, Algebras de Łukasiewicz n×m-valuadas con negación, Ph.D. thesis, Universidad Nacional del Sur, Argentina (2004).

[13] [13] C. Sanza, Notes on n×m–valued Łukasiewicz algebras with negation, Logic Journal of the IGPL, vol. 12 (2004), pp. 499–507 | DOI

[14] [14] C. Sanza, n×m–valued Łukasiewicz algebras with negation, Reports on Mathematical Logic, vol. 40 (2006), pp. 83–106.

[15] [15] C. Sanza, On n×m–valued Łukasiewicz-Moisil algebras, Central European Journal of Mathematics, vol. 6 (2008), pp. 372–383 | DOI

[16] [16] W. Suchoń, Matrix Łukasiewicz algebras, Reports on Mathematical Logic, vol. 4 (1975), pp. 91–104.