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Equational spectrum of Hilbert varieties

100%
Padmanabhan R., Rudeanu S.
Open Mathematics
|
2009
|
tom 7
|
nr 1
66-72
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.
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A glimpse of deductive systems in algebra

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Buşneag D., Rudeanu S.
Open Mathematics
|
2010
|
tom 8
|
nr 4
688-705
EN
The concept of a deductive system has been intensively studied in algebraic logic, per se and in connection with various types of filters. In this paper we introduce an axiomatization which shows how several resembling theorems that had been separately proved for various algebras of logic can be given unique proofs within this axiomatic framework. We thus recapture theorems already known in the literature, as well as new ones. As a by-product we introduce the class of pre-BCK algebras.
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