n x m Łukasiewicz algebras with parameters

• Autorzy: Sanza C.
• Języki publikacji: EN

Abstrakty

EN

Matrix Łukasiewicz algebras were introduced by W. Suchoń in 1975 (Matrix Łukasiewicz Algebras, Reports on Mathematical Logic 4 (1975), 91-104). In this paper n x m{valued Łukasiewicz algebras with negation (or NSnxm algebras are defined and investigated. These algebras constitute an extension of those given by W. Suchoń and in m = 2 case they coincide with n valued Łukasiewicz algebras. Firstly, some of the main results established for matrix Łukasiewicz algebras are extended to NSnxm algebras. In particular, a functional representation theorem is given. Next, NSnxm congruences are determined by taking into account an implication operation which is defined on these algebras. In addition, it is proved that the class of NSnxm algebras is a variety. Besides, subdirectly irreducible algebras are characterized. As a consequence, it is shown that this variety is semisimple and locally finite. Finally, the algebra which generates the variety of NSnxm algebras is obtained and an equational base for the latter is determined.

Słowa kluczowe

EN

Łukasiewicz algebras

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Reports on Mathematical Logic
• Rocznik: 2006
• Tom: Nr 40
• Strony: 83--106
• Opis fizyczny: Bibliogr. 24 poz., rys.

Twórcy

autor

Sanza C.

• Departamento de Matematica. Universidad Nacional del Sur. 8000 Bahia Blanca. Argentina,

Bibliografia

• [1] R. Balbes and Ph. Dwinger, Distributive Lattices, Univ. of Missouri Press,Columbia, 1974.
• [2] N. Belnap, How a computer should think, Contemporary Aspects of Philosophy, G.Ryle Ed. Oriel Press, Boston (1976), pp. 30-56.
• [3] N. Belnap, A useful four valued logic, Modern Uses of Multiple-Valued Logic, J. M.Dunn and G. Epstein, Eds. Reidel, Dordrecht-Boston (1977), pp. 8-37.
• [4] V. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu, Lukasiewicz - Moisil Alge-bras, Annals of Discrete Mathematics 49, North - Holland, 1991.
• [5] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics 78, Springer-Verlag, Berlin, 1981.
• [6] R. Cignoli, Algebras de Moisil de orden n, Ph.D. Thesis. Univ. Nacional del Sur, Bahía Blanca, Argentina, 1969.
• [7] R. Cignoli, Moisil Algebras, Notas de Lógica Matemática 27, Univ. Nacional del Sur, Bahía Blanca, Argentina, 1970.
• [8] R. Cignoli, Some algebraic aspects of many-valued logics, Brazilian Conference on Mathematical Logic 3 (1980), pp. 49-69.
• [9] R. Cignoli, Proper n{valued Lukasiewicz algebras as S-algebras of Lukasiewicz n-valued propositional calculi, Studia Logica 41 (1982), pp. 3-16.
• [10] R. Cignoli, I. D'Ottaviano and D. Mundici, Algebras das Logicas de Lukasiewicz,Campinas UNICAMP - CLE, 1995.
• [11] R. Cignoli, I. D'Ottaviano and D. Mundici, Algebraic Foundations of Many-valued Reasoning, Kluwer, 2000.
• [12] A. V. Figallo, On the congruence in four{valued modal algebras, Portugaliae Math. 49 (1992), pp. 249{261.
• [13] A. V. Figallo, Topicós sobre álgebras modales 4-valuadas, Proceeding of the IX Simposio Latino{Americano de Logica Matemática, (Bahía Blanca, Argentina, 1992), Ed. vol. 39 de Notas de Lógica Matemática, pp. 145-157.
• [14] J. Font and M. Rius, An abstract algebraic logic approach to tretavalent modal logics,Journal of Symbolic Logic 65 (2000), pp. 481-518.
• [15] J. Kalman, Lattice with involution, Trans. Amer. Math. Soc., 87 (1958), pp. 485-491.
• [16] I. Loureiro, Axiomatisation et propriétes des algébres modales tétravalentes, C.R.Acad. Sc. Paris t. 295 (1982), Série I, pp. 555-557.
• [17] I. Loureiro, Algebras Modais Tetravalentes, Ph. D. Thesis, Faculdade de Ciências de Lisboa, 1983.
• [18] G. Moisil, Notes sur les logiques non{chrysippiennes, Ann. Sci. Univ. Jassy, 27 (1941), pp. 86-98.
• [19] G. Moisil, Le algébre di Łukasiewicz, Acta Logica, Bucharest, 6 (1963), pp. 97-135.
• [20] G. Moisil, Sur les logiques de Łukasiewicz a un nombre i de valeurs, Rev. Roum.Math. Pures Appl., 9 (1964), pp. 905-920.
• [21] G. Moisil, Essais sur les logiques non Chrysippiennes, Bucarest, 1972.
• [22] A. Monteiro, Algebras de De Morgan, Curso dictado en la Univ. Nacional del Sur, Bahiá Blanca, Argentina, 1962.
• [23] L. Monteiro, M-algébres de Kleene, An. Acad. brasil. Ciênc. 53, 4 (1981), pp. 665-672.
• [24] W. Suchoń, Matrix Łukasiewicz Algebras, Reports on Mathematical Logic, 4 (1975), pp. 91-104.