n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM n×m -algebra is computed.
@article{bwmeta1.element.doi-10_2478_s11533-008-0035-7,
author = {Claudia Sanza},
title = {On n $\times$ m-valued \L ukasiewicz-Moisil algebras},
journal = {Open Mathematics},
volume = {6},
year = {2008},
pages = {372-383},
zbl = {1155.06009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0035-7}
}
Claudia Sanza. On n × m-valued Łukasiewicz-Moisil algebras. Open Mathematics, Tome 6 (2008) pp. 372-383. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0035-7/
[1] Balbes R., Dwinger P., Distributive lattices, University of Missouri Press, Columbia, Mo., 1974 | Zbl 0321.06012
[2] Boicescu V., Filipoiu A., Georgescu G., Rudeanu S., Łukasiewicz–Moisil algebras, Annals of Discrete Mathematics, 49, North-Holland Publishing Co., Amsterdam, 1991 | Zbl 0726.06007
[3] Bulman-Fleming S., Werner H., Equational compactness in quasi-primal varieties, Algebra Universalis, 1977, 7, 33–46 http://dx.doi.org/10.1007/BF02485416 | Zbl 0367.08006
[4] Burris S., Sankappanavar H.P., A course in universal algebra, Graduate Texts in Mathematics, 78, Springer-Verlag, New York-Berlin, 1981 | Zbl 0478.08001
[5] Cignoli R., Moisil algebras, Notas de Lógica Matemática, 27, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca, 1970
[6] Cignoli R., Proper n-valued Łukasiewicz algebras as S-algebras of Łukasiewicz n-valued propositional calculi, Studia Logica, 1982, 41, 3–16 http://dx.doi.org/10.1007/BF00373490 | Zbl 0509.03012
[7] Cignoli R., D’Ottaviano I., Mundici D., Algebras das logicas de Łukasiewicz, Coleção CLE, 12, Campinas UNICAMP-CLE, 1995
[8] Cignoli R., D’Ottaviano I., Mundici D., Algebraic foundations of many-valued reasoning, Trends in Logic-Studia Logica Library, 7, Kluwer Academic Publishers, Dordrecht, 2000
[9] Cornish W., Fowler P., Coproducts of De Morgan algebras, Bull. Austral. Math. Soc., 1977, 16, 1–13 http://dx.doi.org/10.1017/S0004972700022966 | Zbl 0329.06005
[10] Figallo A.V., Pascual I., Ziliani A., Notes on monadic n-valued Łukasiewicz algebras, Math. Bohem., 2004, 129, 255–271 | Zbl 1080.06011
[11] Grigolia R., Algebraic analysis of Łukasiewicz-Tarski’s n-valued logical systems, In: Wójcicki R., Malinowski G. (Eds.), Selected papers on Łukasiewicz sentential calculi, Zaklad Narod. im. Ossolin., Wydawn. Polsk. Akad. Nauk, Wroclaw, 1977, 81–92
[12] Iorgulescu A., Connections between MVn algebras and n-valued Łukasiewicz-Moisil algebras IV, Journal of Universal Computer Science, 2000, 6, 139–154
[13] Łukasiewicz J., On three-valued logic, Ruch Filozoficzny, 1920, 5, 160–171
[14] Moisil Gr.C., Notes sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy, 1941, 27, 86–98
[15] Moisil Gr.C., Essais sur les logiques non chrysippiennes, Éditions de l’Académie de la République Socialiste de Roumanie, Bucharest, 1972
[16] Post E., Introduction to a general theory of elementary propositions, Amer. J. Math., 1921, 43, 163–185 http://dx.doi.org/10.2307/2370324 | Zbl 48.1122.01
[17] Priestley H., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc., 1970, 2, 186–190 http://dx.doi.org/10.1112/blms/2.2.186 | Zbl 0201.01802
[18] Priestley H., Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc., 1972, 4, 507–530 http://dx.doi.org/10.1112/plms/s3-24.3.507 | Zbl 0323.06011
[19] Priestley H., Ordered sets and duality for distributive lattices, Ann. Discrete Math., North-Holland, Amsterdam, 1984, 23, 39–60
[20] Sanza C., Algebras de Łukasiewicz matriciales n × m-valuadas con negación, Noticiero de la Unión Matemática Argentina, Rosario (Argentina), 2000
[21] Sanza C., Algebras de Łukasiewicz n × m-valuadas con negación, Ph.D. thesis, Universidad Nacional del Sur. Argentina, 2004
[22] Sanza C., Notes on n × m-valued Łukasiewicz algebras with negation, Log. J. IGPL, 2004, 12, 499–507 http://dx.doi.org/10.1093/jigpal/12.6.499 | Zbl 1062.06018
[23] Sanza C., n × m-valued Łukasiewicz algebras with negation, Rep. Math. Logic, 2006, 40, 83–106
[24] Suchoń W., Matrix Łukasiewicz Algebras, Rep. Math. Logic, 1975, 4, 91–104
[25] Tarski A., Logic semantics metamathematics, Clarendon Press, Oxford, 1956
[26] Werner H., Discriminator-algebras, Algebraic representation and model theoretic properties, Akademie-Verlag, Berlin, 1978 | Zbl 0374.08002