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1
Content available remote Semitopological BL-algebras and MV-algebras
Zahiri O., Borzooei R. A.
Demonstratio Mathematica
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2014
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Vol. 47, nr 3
522--538
EN
In this paper, by considering the notion of upsets, for any element x of a BL-algebra L, we construct a topology γx on L and show that L-algebras with this topology formes a semitopological BL-algebras. Then we obtain some of the topological aspects of this structure such as connectivity and compactness. Moreover, we introduced two kinds of semitopological MV-algebra by using two kinds of definition of MV–algebra and show that they are equivalent.
2
Content available remote On implicative and maximal ideals of BL-algebras
Walendziak A.
Commentationes Mathematicae
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2014
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Vol 54, No. 2
247--258
EN
In the theory of MV-algebras, implicative ideals are studied by Hoo and Sessa. In this paper we define and characterize implicative ideals of BL-algebras. We also investigate maximal ideals of BL-algebras and prove that if an ideal is prime and implicative, then it is maximal. Moreover, we show that an ideal is maximal if and only if the quotient BL-algebra is a simple MV-algebra. Finally, we give the homomorphic properties of implicative and maximal ideals.
3
Content available remote When is a BCC-algebra equivalent to an MV-algebra?
Chajda I., Halas R., Kuhr J.
Demonstratio Mathematica
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2007
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Vol. 40, nr 4
759-768
EN
The aim of this paper is to characterize BCC-algebras which are term equivalent to MV-algebras. It turns out that they arę just the bounded commutative BCC-algebras. Purther, we characterize congruence kernels as deductive systems. The explicit description of a principal deductive system enables us to prove that every subdirectly irreducible bounded commutatwe BCC-algebra is a chain (with respect to the induced order) .
4
Content available remote Mv-like algebras associated to lambda-ortholattices
Chajda I., Emanovsky P., Kolarik M.
Demonstratio Mathematica
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2007
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Vol. 40, nr 2
261-270
EN
The concept of a ambda-lattice generalizes a lattice by substituting associativity by the so-called skew associativity. When a bounded ambda-lattice is equipped with a monotonous unary involution which is a complementation, it is called a ambda-ortholattice. For ambda-ortholattices a Sheffer operation is constructed and, moreover, a derived algebra analogous to an MV-algebra is assigned whenever the ambda-lattice has antitone involutions on sections.
5
Content available remote Modifications of MV-algebras corresponding to strong ortholattices
Chajda I., Länger H.
Demonstratio Mathematica
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2005
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Vol. 38, nr 1
1--6
EN
It is well-known that principal filters of MV-algebras are de Morgan algebras with involutory complementation. A modification of the notion of an MV-algebra is presented having the property that all principal filters are ortholattices. It turns out that the commutativity of these modified MV-algebras is equivalent to the distributivity of the corresponding ortholattices.
6
Content available remote On pseudo-effect algebras which can be covered by pseudo MV-algebras
Dvurečenskij A., Vetterlein T.
Demonstratio Mathematica
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2003
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Vol. 36, nr 2
261--282
EN
Pseudo-effect algebras are partial algebras (E; +, 0,1) which were recently introduced. They have a partially defined addition + which is only associative and not necessary commutative and with two complements, left and right ones. They are a non-commutative generalization of orthomodular posets and MV-algebras, respectively. We define five kinds of compatibilities, and we introduce a block as a maximal set of mutually compatible elements. The compatibility is a property of the physical system which corresponds to the distributivity, or equivalently, to "classical mechanics"-type phenomena. We show that any lattice pseudo-effect algebra under a natural condition can be covered by blocks, and any block is a pseudo MV-algebra. This result generalizes the analogical result of Riecanova for effect algebras. If the pseudo-effect algebra with the condition is, in addition, a (7-complete lattice, then it is a commutative effect algebra which can be covered by cr-complete MV-algebras.
7
Content available remote Distributive atomic efect algebras
Riečanová Z.
Demonstratio Mathematica
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2003
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Vol. 36, nr 2
247--259
EN
Motivated by the use of fuzzy or unsharp quantum logics as carriers of probability measures there have been recently introduced effect algebras (D-posets). We extend a result by Greechie, Foulis and Pulmannova of finite distributive effect algebras to all Archimedean atomic distributive effect algebras. We show that every such an effect algebra is join and meet dense in a complete effect algebra being a direct product of finite chains and distributive diamonds. This proves that every such effect algebra has a MacNeille completion being again a distributive effect algebra and both these effect algebras are continuous lattices. Moreover, we show that every faithful or (o)-continuous state (probability) on such an effect algebra is a valuation, hence a subadditive state. Its existence is also proved. Finally, we prove that every complete atomic distributive effect algebra E is a homomorphic image of a complete modular atomic ortholattice regarded as effect algebra and E is an MV-effect algebra (MV-algebra) if and only if it is a homomorphic image of a Boolean algebra regarded as effect algebra.
8
Content available remote Regularity of generalized MV-algebras
Chajda I., Dorfer G., Halaš R.
Demonstratio Mathematica
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2001
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Vol. 34, nr 1
25-32
EN
A generalized MV algebra is constructed by means of an 1-group m a way similar to that an MV-algebra is related to a commutative 1-group, see e.g. [11], [9]. We prove that the variety of generalized MV-algebras is congruence regular and give an explicit description of congruence classes.
9
Content available remote Orthogonal sets in efect algebras
Riecanova Z.
Demonstratio Mathematica
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2001
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Vol. 34, nr 3
525-532
EN
We show that for a lattice effect algebra two conceptions of completeness (o-completeness) coincide. Moreover, a separable effect algebra is complete if and only if it is cr-complete. Further, in an Archimedean atomic lattice effect algebra to every nonzero element x there is a ^-orthogonal system G of not necessary different atoms such that x = G. A lattice effect algebra E is complete if and only if every block of E is complete. Every atomic Archimedean lattice effect algebra is a union of atomic blocks, since each of its elements is a sum of a ^-orthogonal system of atoms.
10
Content available remote Ideals and congruences in generalized MV-algebras
Chajda I., Halas R., Rachunek J.
Demonstratio Mathematica
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2000
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Vol. 33, nr 2
213-222
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