Abstract
\({{\mathcal {M}} L}^{m}_n\) of monadic m-generalized Łukasiewicz algebras of order n (or \(M L^{m}_n\)-algebras), namely a generalization of monadic n-valued Łukasiewicz algebras. In this article, we determine the congruences and we characterized the subdirectly irreducible \(M L^{m}_n\)-algebras. From this last result we proved that \({{\mathcal {M}} L}^{m}_n\) is a discriminator variety and as a consequence we characterized the principal congruences. In the last part of this paper we find an immersion of these algebras in a functional algebra and we proved that in the finite case they are isomorphic. This last result allows to show a new functional representation for monadic n-valued Łukasiewicz algebras. Finally, we define the notions of \(M L^{m}_n\)-algebra of fractions and maximal algebra of fractions and we prove the existence of a maximal \(ML^{m}_n\)-algebra of fractions for an \(ML^{m}_n\)-algebra.
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References
Almada, T., and J. Vaz-De-Carvalho, A generalization of the Łukasiewicz algebras, Studia Logica 69:329–338, 2001.
Ashraf, M., and A. Shakir, On left multipliers and the commutativity of prime rings, Demonstratio Mathematica 41:761–761, 2008.
Balbes, R., and Ph. Dwinger, Distributive Lattices, Univ. of Missouri Press, Columbia, 1974.
Boicescu, V., A. Filipoiu, G. Georgescu, and S. Rudeanu, Łukasiewicz-Moisil Algebras, vol. 49 of Annals of Discrete Mathematics, North-Holland, 1991.
Busneag, D., and F. Chirtes, \(LM_n\)-algebra of fractions and maximal \(LM_n\)-algebra of fractions, Discrete Mathematics 296: 143–165, 2005.
Cignoli, R., Moisil Algebras, vol. 27 of Notas de Lógica Matemática, Univ. Nacional del Sur, Bahía Blanca, 1970.
Cirulis, J., Multipliers in implicative algebras, Bulletin of the Section of Logic 15: 152–157, 1986.
Colville, P., G. Davis, and K. Keimel, Positive Derivations on f-rings, Journal of the Australian Mathematical Society 23: 371–375, 1977.
Cornish, W. H., The multiplier extension of a distributive lattice, Journal of Algebra 32: 339–355, 1974.
Cornish, W. H., A multiplier approach to implicative BCK-algebras, Mathematics Seminar Notes 8: 157–169, 1980.
Figallo, A. V., C. Gallardo, and A. Ziliani, Weak implication on generalized Łukasiewicz algebras of order n, Bulletin of the Section of Logic 39: 187–198, 2010.
Figallo, A. V., I. Pascual and A. Ziliani, Notes on monadic n-valued Łukasiewicz algebras, Mathematica Bohemica 129: 255–271, 2004.
Figallo, A. V., C. Sanza and A. Ziliani, Functional monadic n-valued Łukasiewicz algebras, Mathematica Bohemica 130: 337–348, 2005.
Gallardo, C., Sobre las álgebras de Łukasiewicz\(m-\)generalizadas de orden\(n\), Ph. D. Thesis, Univ. Nac. del Sur, Bahía Blanca, 2016.
Gallardo, C., and A. Ziliani, The \(L^{m}_{n}-\)propositional calculus, Mathematica Bohemica, 140: 11–33, 2015.
Gallardo, C., and A. Ziliani, Localization of m-generalized Łukasiewicz algebras of order n, Journal of Multiple-Valued Logic and Soft Computing 27: 21–45, 2016.
Georgescu, G., and C. Vraciu, Algebre Boole monadice si algebre Łukasiewicz monadice, Studii Cercet. Mat. 23: 1025–1048, 1971.
Halmos, P. R. Algebraic Logic, Chelsea, New York, 1962.
Iorgulescu, A., On the construction of three-valued Łukasiewicz-Moisil algebras, Discrete Mathematics 48: 213–227, 1984.
Iorgulescu, A., On BCK Algebras - Part I.a. An Attempt to Treat Unitarily the Algebras of Logic. New Algebras, Journal of Universal Computer Science 13: 1628–1654, 2007.
Iorgulescu, A., On BCK Algebras - Part I.b. An Attempt to Treat Unitarily the Algebras of Logic. New Algebras, Journal of Universal Computer Science 14: 3686–3715, 2008.
Iorgulescu, A., Algebras of logic as BCK–algebras, Academy of Economic Studies Press, Bucharest, 2008.
Kolibiar, M., Bermerkungen über Translationen der Verbände, Acta Fac. Rerum Nat. Univ. Comenian Math. 5: 455–458, 1961.
Laca, M., and I. Raeburn, Extending Multipliers from Semigroups, Proceedings of the American Mathematical Society 123: 355–362, 1995.
Lambek, J., Lectures on Rings and Modules, Blaisdell Publishing Company, 1966.
Petrich, M., The translational hull in semigroups and rings, Semigroup Forum 1: 283–360, 1970.
Rudeanu, S., Localizations and fractions in algebra of logic, Journal of Multiple-Valued Logic and Soft Computing 16: 467–504, 2010.
Schmid, J., Multipliers on distributive lattices and rings of fractions, Houston Journal of Mathematics 6: 401–425, 1980.
Szász, G., Die Translationen der Halbverbände, Acta Scientiarum Mathematicarum 17: 165–169, 1956.
Szász, G., and J. Szendrei, Über der Translationen der Halbverbände, Acta Scientiarum Mathematicarum 18: 44–47, 1957.
Vaz De Carvalho, J, On the variety of \(m\)-generalized Łukasiewicz algebras of order \(n\), Studia Logica 94: 291–305, 2010.
Yon, Y. H., and K. H. Kim, Multipliers in subtraction algebras, Scientiae Mathematicae Japonicae 73: 117–123, 2011.
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The authors are truly thankful to the referee for several helpful suggestions for improvements in this paper.
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Gallardo, C., Ziliani, A. A Generalization of Monadic n-Valued Łukasiewicz Algebras. Stud Logica 110, 457–478 (2022). https://doi.org/10.1007/s11225-021-09968-9
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DOI: https://doi.org/10.1007/s11225-021-09968-9