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2009 | 7 | 1 | 66-72
Tytuł artykułu

Equational spectrum of Hilbert varieties

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.
Wydawca
Czasopismo
Rocznik
Tom
7
Numer
1
Strony
66-72
Opis fizyczny
Daty
wydano
2009-03-01
online
2009-01-10
Twórcy
  • Faculty of Mathematics and Informatics, University of Bucharest, Bucharest, Romania, srudeanu@yahoo.com
Bibliografia
  • [1] Abbott J.C., Implicational algebras, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 1967, 11(59), 3–23
  • [2] Abbott J.C., Semi-Boolean algebra, Mat. Vesnik, 1967, 4(19), 177–198
  • [3] Buşneag D., Contribuţii la studiul algebrelor Hilbert, PhD thesis, University of Bucharest, Romania, 1985 (in Romanian)
  • [4] Buşneag D., Categories of algebraic logic, Editura Academiei Romane, Bucharest, 2006
  • [5] Cornish W.H., Two-based definitions of bounded commutative BCK-algebras, Math. Sem. Notes Kobe Univ., 1983, 11(1), 9–15
  • [6] Diego A., Sobre algebras de Hilbert, PhD thesis, University of Buenos Aires, 1961
  • [7] Diego A., Sur les algèbres de Hilbert, Collection de Logique Mathématique, Sér. A, Fasc. XXI Gauthier-Villars, Paris, E. Nauwelaerts, Louvain, 1966
  • [8] Henkin L., An algebraic characterization of quantifiers, Fund. Math., 1950, 37, 63–74
  • [9] Iorgulescu A., Algebras of logic as BCK-algebras, ASE, Bucharest, 2008
  • [10] McNulty G., Minimum bases for equational theories of groups and rings, Ann. Pure Appl. Logic, 2004, 127, 131–153 http://dx.doi.org/10.1016/j.apal.2003.11.012[Crossref]
  • [11] Padmanabhan R., Rudeanu S., Axioms for lattices and Boolean algebras, World Scientific, Singapore, 2008
  • [12] Pałasiński M., Wożniakowska B., An equational basis for commutative BCK-algebras, Math. Sem. Notes Kobe Univ., 1982, 10, 175–178
  • [13] Rasiowa H., An algebraic approach to non-classical logics, Studies in Logic and the Foundations of Mathematics 78, North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1974
  • [14] Tarski A., Equational logic and equational theories of algebras, Contributions to Math. Logic (Colloquium, Hannover, 1966), North-Holland, Amsterdam 1968, 275–288
  • [15] Yutani H., The class of commutative BCK-algebras is equationally definable, Math. Sem. Notes Kobe Univ., 1977, 5, 207–210
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-008-0060-6
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