On n × m-valued Łukasiewicz-Moisil algebras

Claudia Sanza

Open Mathematics (2008)

  • Volume: 6, Issue: 3, page 372-383
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Claudia Sanza. "On n × m-valued Łukasiewicz-Moisil algebras." Open Mathematics 6.3 (2008): 372-383. <http://eudml.org/doc/269521>.

@article{ClaudiaSanza2008,
abstract = {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.},
author = {Claudia Sanza},
journal = {Open Mathematics},
keywords = {n-valued Łukasiewicz-Moisil algebras; Priestley spaces; discriminator varieties; congruences},
language = {eng},
number = {3},
pages = {372-383},
title = {On n × m-valued Łukasiewicz-Moisil algebras},
url = {http://eudml.org/doc/269521},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Claudia Sanza
TI - On n × m-valued Łukasiewicz-Moisil algebras
JO - Open Mathematics
PY - 2008
VL - 6
IS - 3
SP - 372
EP - 383
AB - 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.
LA - eng
KW - n-valued Łukasiewicz-Moisil algebras; Priestley spaces; discriminator varieties; congruences
UR - http://eudml.org/doc/269521
ER -

References

top
  1. [1] Balbes R., Dwinger P., Distributive lattices, University of Missouri Press, Columbia, Mo., 1974 Zbl0321.06012
  2. [2] Boicescu V., Filipoiu A., Georgescu G., Rudeanu S., Łukasiewicz–Moisil algebras, Annals of Discrete Mathematics, 49, North-Holland Publishing Co., Amsterdam, 1991 Zbl0726.06007
  3. [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 Zbl0367.08006
  4. [4] Burris S., Sankappanavar H.P., A course in universal algebra, Graduate Texts in Mathematics, 78, Springer-Verlag, New York-Berlin, 1981 Zbl0478.08001
  5. [5] Cignoli R., Moisil algebras, Notas de Lógica Matemática, 27, Instituto de Matemática, Universidad Nacional del Sur, Bahía Blanca, 1970 
  6. [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 Zbl0509.03012
  7. [7] Cignoli R., D’Ottaviano I., Mundici D., Algebras das logicas de Łukasiewicz, Coleção CLE, 12, Campinas UNICAMP-CLE, 1995 
  8. [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. [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 Zbl0329.06005
  10. [10] Figallo A.V., Pascual I., Ziliani A., Notes on monadic n-valued Łukasiewicz algebras, Math. Bohem., 2004, 129, 255–271 Zbl1080.06011
  11. [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. [12] Iorgulescu A., Connections between MVn algebras and n-valued Łukasiewicz-Moisil algebras IV, Journal of Universal Computer Science, 2000, 6, 139–154 
  13. [13] Łukasiewicz J., On three-valued logic, Ruch Filozoficzny, 1920, 5, 160–171 
  14. [14] Moisil Gr.C., Notes sur les logiques non-chrysippiennes, Ann. Sci. Univ. Jassy, 1941, 27, 86–98 
  15. [15] Moisil Gr.C., Essais sur les logiques non chrysippiennes, Éditions de l’Académie de la République Socialiste de Roumanie, Bucharest, 1972 
  16. [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 Zbl48.1122.01
  17. [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 Zbl0201.01802
  18. [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 Zbl0323.06011
  19. [19] Priestley H., Ordered sets and duality for distributive lattices, Ann. Discrete Math., North-Holland, Amsterdam, 1984, 23, 39–60 
  20. [20] Sanza C., Algebras de Łukasiewicz matriciales n × m-valuadas con negación, Noticiero de la Unión Matemática Argentina, Rosario (Argentina), 2000 
  21. [21] Sanza C., Algebras de Łukasiewicz n × m-valuadas con negación, Ph.D. thesis, Universidad Nacional del Sur. Argentina, 2004 
  22. [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 Zbl1062.06018
  23. [23] Sanza C., n × m-valued Łukasiewicz algebras with negation, Rep. Math. Logic, 2006, 40, 83–106 
  24. [24] Suchoń W., Matrix Łukasiewicz Algebras, Rep. Math. Logic, 1975, 4, 91–104 
  25. [25] Tarski A., Logic semantics metamathematics, Clarendon Press, Oxford, 1956 
  26. [26] Werner H., Discriminator-algebras, Algebraic representation and model theoretic properties, Akademie-Verlag, Berlin, 1978 Zbl0374.08002

NotesEmbed ?

top

You must be logged in to post comments.