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Relatively pseudocomplemented Hilbert algebras

100%
Cīrulis J.
Open Mathematics
|
2008
|
tom 6
|
nr 1
189-190
EN
We characterise those Hilbert algebras that are relatively pseudocomplemented posets.
2
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Hilbert algebras as implicative partial semilattices

75%
Cīrulis J.
Open Mathematics
|
2007
|
tom 5
|
nr 2
264-279
EN
The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.
3
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Equational spectrum of Hilbert varieties

75%
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.
4
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A glimpse of deductive systems in algebra

75%
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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