Equational spectrum of Hilbert varieties

R. Padmanabhan; Sergiu Rudeanu

Open Mathematics (2009)

  • Volume: 7, Issue: 1, page 66-72
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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R. Padmanabhan, and Sergiu Rudeanu. "Equational spectrum of Hilbert varieties." Open Mathematics 7.1 (2009): 66-72. <http://eudml.org/doc/269360>.

@article{R2009,
abstract = {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.},
author = {R. Padmanabhan, Sergiu Rudeanu},
journal = {Open Mathematics},
keywords = {Hilbert algebra; Implication algebra; BCK algebra; Equational spectrum; Equational class; One-based theory; Boolean algebra; implication algebra; Tarski algebra; BCK-algebra; equational spectrum; equational class; subvariety},
language = {eng},
number = {1},
pages = {66-72},
title = {Equational spectrum of Hilbert varieties},
url = {http://eudml.org/doc/269360},
volume = {7},
year = {2009},
}

TY - JOUR
AU - R. Padmanabhan
AU - Sergiu Rudeanu
TI - Equational spectrum of Hilbert varieties
JO - Open Mathematics
PY - 2009
VL - 7
IS - 1
SP - 66
EP - 72
AB - 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.
LA - eng
KW - Hilbert algebra; Implication algebra; BCK algebra; Equational spectrum; Equational class; One-based theory; Boolean algebra; implication algebra; Tarski algebra; BCK-algebra; equational spectrum; equational class; subvariety
UR - http://eudml.org/doc/269360
ER -

References

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  10. [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] Zbl1049.03027
  11. [11] Padmanabhan R., Rudeanu S., Axioms for lattices and Boolean algebras, World Scientific, Singapore, 2008 Zbl1158.06001
  12. [12] Pałasiński M., Wożniakowska B., An equational basis for commutative BCK-algebras, Math. Sem. Notes Kobe Univ., 1982, 10, 175–178 Zbl0504.03029
  13. [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. [14] Tarski A., Equational logic and equational theories of algebras, Contributions to Math. Logic (Colloquium, Hannover, 1966), North-Holland, Amsterdam 1968, 275–288 
  15. [15] Yutani H., The class of commutative BCK-algebras is equationally definable, Math. Sem. Notes Kobe Univ., 1977, 5, 207–210 Zbl0373.02045

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