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.
@article{bwmeta1.element.doi-10_2478_s11533-008-0060-6,
author = {R. Padmanabhan and Sergiu Rudeanu},
title = {Equational spectrum of Hilbert varieties},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {66-72},
zbl = {1209.03056},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0060-6}
}
R. Padmanabhan; Sergiu Rudeanu. Equational spectrum of Hilbert varieties. Open Mathematics, Tome 7 (2009) pp. 66-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-008-0060-6/
[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 | Zbl 05191994
[5] Cornish W.H., Two-based definitions of bounded commutative BCK-algebras, Math. Sem. Notes Kobe Univ., 1983, 11(1), 9–15 | Zbl 0553.03044
[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 | Zbl 0041.34804
[9] Iorgulescu A., Algebras of logic as BCK-algebras, ASE, Bucharest, 2008 | Zbl 1172.03038
[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] | Zbl 1049.03027
[11] Padmanabhan R., Rudeanu S., Axioms for lattices and Boolean algebras, World Scientific, Singapore, 2008 | Zbl 1158.06001
[12] Pałasiński M., Wożniakowska B., An equational basis for commutative BCK-algebras, Math. Sem. Notes Kobe Univ., 1982, 10, 175–178 | Zbl 0504.03029
[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 | Zbl 0373.02045