Equational spectrum of Hilbert varieties

R. Padmanabhan 1  and Sergiu Rudeanu 2
  • 1 Department of Mathematics, University of Manitoba, Winnipeg, Canada
  • 2 Faculty of Mathematics and Informatics, University of Bucharest, Bucharest, Romania

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.

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