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Hilbert algebras as implicative partial semilattices

100%
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.
2
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Strong ideals and horizontal ideals in pseudo-BCH-algebras

100%
Walendziak A.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2016
|
tom 15
EN
In this paper we define strong ideals and horizontal ideals in pseudo-BCH-algebras and investigate the properties and characterizations of them.
3
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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.
4
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Topological representation for monadic implication algebras

76%
Abad M., Cimadamore C., Díaz Varela J.
Open Mathematics
|
2009
|
tom 7
|
nr 2
299-309
EN
In this paper, every monadic implication algebra is represented as a union of a unique family of monadic filters of a suitable monadic Boolean algebra. Inspired by this representation, we introduce the notion of a monadic implication space, we give a topological representation for monadic implication algebras and we prove a dual equivalence between the category of monadic implication algebras and the category of monadic implication spaces.
5
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The Sheffer stroke operation reducts of basic algebras

76%
Oner T., Senturk I.
Open Mathematics
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2017
|
tom 15
|
nr 1
926-935
EN
In this study, a term operation Sheffer stroke is presented in a given basic algebra 𝒜 and the properties of the Sheffer stroke reduct of 𝒜 are examined. In addition, we qualify such Sheffer stroke basic algebras. Finally, we construct a bridge between Sheffer stroke basic algebras and Boolean algebras.
6
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Equational spectrum of Hilbert varieties

76%
Padmanabhan R., Rudeanu S.
Open Mathematics
|
2009
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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.
7
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Some properties of epimorphisms of Hilbert algebras

76%
Buşneag D., Ghiţă M.
Open Mathematics
|
2010
|
tom 8
|
nr 1
41-52
EN
This paper represents a start in the study of epimorphisms in some categories of Hilbert algebras. Even if we give a complete characterization for such epimorphisms only for implication algebras, the following results will make possible the construction of some examples of epimorphisms which are not surjective functions. Also, we will show that the study of epimorphisms of Hilbert algebras is equivalent with the study of epimorphisms of Hertz algebras.
8
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Representation and duality for Hilbert algebras

76%
Celani S., Cabrer L., Montangie D.
Open Mathematics
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2009
|
tom 7
|
nr 3
463-478
EN
In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.
9
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Algebraic axiomatization of tense intuitionistic logic

64%
Chajda I.
Open Mathematics
|
2011
|
tom 9
|
nr 5
1185-1191
EN
We introduce two unary operators G and H on a relatively pseudocomplemented lattice which form an algebraic axiomatization of the tense quantifiers “it is always going to be the case that” and “it has always been the case that”. Their axiomatization is an extended version for the classical logic and it is in accordance with these operators on many-valued Łukasiewicz logic. Finally, we get a general construction of these tense operators on complete relatively pseudocomplemented lattice which is a power lattice via the so-called frame.
10
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Quantum B-algebras

52%
Rump W.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1881-1899
EN
The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.
11
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A glimpse of deductive systems in algebra

52%
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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