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

• Autorzy: Carlos Gallardo , Alicia Ziliani

Abstrakty

EN

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 \(\mathbf{NS}_{3\times 3}\).  

Słowa kluczowe

EN

\(n\)-valued Łukasiewicz-Moisil algebras; \(n \times m\)-valued Łukasiewicz algebras with negation; free algebras; lattice of subvarieties

• Wydawca: Wydawnictwo Uniwersytetu Łódzkiego
• Czasopismo: Bulletin of the Section of Logic
• Rocznik: 2021
• Tom: 50
• Numer: 3
• Strony: 289-298

Daty

wydano

2021

Twórcy

autor

Carlos Gallardo

• Universidad Nacional del Sur, Departamento de Matemática

autor

Alicia Ziliani

• Universidad Nacional del Sur, Departamento de Matemática

• https://orcid.org/0000000229085218

Bibliografia

• [1] N. Belnap, How a computer should think, Oriel Press, Boston (1977), pp. 30–56.
• [2] V. Boicescu, A. Filipoiu, G. Georgescu, S.Rudeanu, Łukasiewicz-Moisil Algebras, vol. 49 of Annals of Discrete Mathematics, North-Holland, Amsterdam (1991).
• [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] R. Cignoli, Moisil Algebras, vol. 27 of Notas de Lógica Matemática, Universidad Nacional del Sur, Argentina (1970).
• [5] A. Day, Splitting algebras and a weak notion of projectivity, Algebra Universalis, vol. 5 (1975), pp. 153–162, DOI: https://doi.org/10.1007/BF02485249
• [6] A. Day, On the lattice of subvarietes, Houston Journal of Mathematics, vol. 5 (1979), pp. 183–192.
• [7] A. V. Figallo, C. Sanza, The NSn×m–propositional calculus, Bulletin of the Section of Logic, vol. 37 (2008), pp. 67–79.
• [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: https://doi.org/10.2478/auom-2013-0019
• [9] B. Jonnson, Algebras whose congruence lattices are distributive, Mathematica Scandinavica, vol. 21 (1967), pp. 110–121, DOI: https://doi.org/10.7146/math.scand.a-10850
• [10] G. Moisil, Notes sur les logiques non-chrysippiennes, Annales Scientifiques de l’Université de Jassy, vol. 27 (1941), pp. 86–98.
• [11] G. Moisil, Le algebre di Łukasiewicz, Analele Universitii Bucureti, seria Acta logica, vol. 6 (1963), pp. 97–135.
• [12] C. Sanza, Algebras de Łukasiewicz n×m-valuadas con negación, Ph.D. thesis, Universidad Nacional del Sur, Argentina (2004).
• [13] C. Sanza, Notes on n×m–valued Łukasiewicz algebras with negation, Logic Journal of the IGPL, vol. 12 (2004), pp. 499–507, DOI: https://doi.org/10.1093/jigpal/12.6.499
• [14] C. Sanza, n×m–valued Łukasiewicz algebras with negation, Reports on Mathematical Logic, vol. 40 (2006), pp. 83–106.
• [15] C. Sanza, On n×m–valued Łukasiewicz-Moisil algebras, Central European Journal of Mathematics, vol. 6 (2008), pp. 372–383, DOI: https://doi.org/10.2478/s11533-008-0035-7
• [16] W. Suchoń, Matrix Łukasiewicz algebras, Reports on Mathematical Logic, vol. 4 (1975), pp. 91–104.

Identyfikatory

DOI

10.18778/0138-0680.2021.10

nonstd.3pn.id

2033855