18 found
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  1.  29
    State-Morphism MV-Algebras.Antonio Di Nola & Anatolij Dvurečenskij - 2009 - Annals of Pure and Applied Logic 161 (2):161-173.
    We present a stronger variation of state MV-algebras, recently presented by T. Flaminio and F. Montagna, which we call state-morphism MV-algebras. Such structures are MV-algebras with an internal notion, a state-morphism operator. We describe the categorical equivalences of such state MV-algebras with the category of unital Abelian ℓ-groups with a fixed state operator and present their basic properties. In addition, in contrast to state MV-algebras, we are able to describe all subdirectly irreducible state-morphism MV-algebras.
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  2.  29
    States on Pseudo MV-Algebras.Anatolij Dvurečenskij - 2001 - Studia Logica 68 (3):301-327.
    Pseudo MV-algebras are a non-commutative extension of MV-algebras introduced recently by Georgescu and Iorgulescu. We introduce states (finitely additive probability measures) on pseudo MV-algebras. We show that extremal states correspond to normal maximal ideals. We give an example in that, in contrast to classical MV-algebras introduced by Chang, states can fail on pseudo MV-algebras. We prove that representable and normal-valued pseudo MV-algebras admit at least one state.
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  3.  12
    Subdirectly Irreducible State-Morphism BL-Algebras.Anatolij Dvurečenskij - 2011 - Archive for Mathematical Logic 50 (1-2):145-160.
    Recently Flaminio and Montagna (Proceedings of the 5th EUSFLAT Conference, II: 201–206. Ostrava, 2007), (Inter. J. Approx. Reason. 50:138–152, 2009) introduced the notion of a state MV-algebra as an MV-algebra with internal state. We have two kinds: state MV-algebras and state-morphism MV-algebras. These notions were also extended for state BL-algebras in (Soft Comput. doi:10.1007/s00500-010-0571-5). In this paper, we completely describe subdirectly irreducible state-morphism BL-algebras and this generalizes an analogous result for state-morphism MV-algebras presented in (Ann. Pure Appl. Logic 161:161–173, 2009).
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  4. Kite Pseudo Effect Algebras.Anatolij Dvurečenskij - 2013 - Foundations of Physics 43 (11):1314-1338.
    We define a new class of pseudo effect algebras, called kite pseudo effect algebras, which is connected with partially ordered groups not necessarily with strong unit. In such a case, starting even with an Abelian po-group, we can obtain a noncommutative pseudo effect algebra. We show how such kite pseudo effect algebras are tied with different types of the Riesz Decomposition Properties. Kites are so-called perfect pseudo effect algebras, and we define conditions when kite pseudo effect algebras have the least (...)
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  5.  25
    On Gleason’s Theorem Without Gleason.David Buhagiar, Emmanuel Chetcuti & Anatolij Dvurečenskij - 2009 - Foundations of Physics 39 (6):550-558.
    The original proof of Gleason’s Theorem is very complicated and therefore, any result that can be derived also without the use of Gleason’s Theorem is welcome both in mathematics and mathematical physics. In this paper we reprove some known results that had originally been proved by the use of Gleason’s Theorem, e.g. that on the quantum logic ℒ(H) of all closed subspaces of a Hilbert space H, dim H≥3, there is no finitely additive state whose range is countably infinite. In (...)
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  6.  15
    On the Structure of Linearly Ordered Pseudo-BCK-Algebras.Anatolij Dvurečenskij & Jan Kühr - 2009 - Archive for Mathematical Logic 48 (8):771-791.
    Pseudo-BCK-algebras are a non-commutative generalization of well-known BCK-algebras. The paper describes a situation when a linearly ordered pseudo-BCK-algebra is an ordinal sum of linearly ordered cone algebras. In addition, we present two identities giving such a possibility of the decomposition and axiomatize the residuation subreducts of representable pseudo-hoops and pseudo-BL-algebras.
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  7.  63
    Piron's and Bell's Geometric Lemmas and Gleason's Theorem.Georges Chevalier, Anatolij Dvurečenskij & Karl Svozil - 2000 - Foundations of Physics 30 (10):1737-1755.
    We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.
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  8.  38
    States on Pseudo Effect Algebras and Integrals.Anatolij Dvurečenskij - 2011 - Foundations of Physics 41 (7):1143-1162.
    We show that every state on an interval pseudo effect algebra E satisfying an appropriate version of the Riesz Decomposition Property (RDP for short) is an integral through a regular Borel probability measure defined on the Borel σ-algebra of a Choquet simplex K. In particular, if E satisfies the strongest type of RDP, the representing Borel probability measure can be uniquely chosen to have its support in the set of the extreme points of K.
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  9.  28
    Atomic Effect Algebras with the Riesz Decomposition Property.Anatolij Dvurečenskij & Yongjian Xie - 2012 - Foundations of Physics 42 (8):1078-1093.
    We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any σ-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states.
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  10.  91
    Bell-Type Inequalities in Horizontal Sums of Boolean Algebras.Anatolij Dvurečenskij & Helmut Länger - 1994 - Foundations of Physics 24 (8):1195-1202.
    We give a necessary and sufficient condition for a Bell-type inequality to hold in a horizontal sum of finitely many finite Boolean algebras.
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  11.  31
    On Categorical Equivalences of Commutative BCK-Algebras.Anatolij Dvurečenskij - 2000 - Studia Logica 64 (1):21-36.
    A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G 0), where G is an Abelian ℓ-group, G 0 is a subset of the positive cone generating G + (...)
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  12.  10
    Perfect Effect Algebras and Spectral Resolutions of Observables.Anatolij Dvurečenskij - 2019 - Foundations of Physics 49 (6):607-628.
    We study perfect effect algebras, that is, effect algebras with the Riesz decomposition property where every element belongs either to its radical or to its co-radical. We define perfect effect algebras with principal radical and we show that the category of such effect algebras is categorically equivalent to the category of unital po-groups with interpolation. We introduce an observable on a \-monotone \-complete perfect effect algebra with principal radical and we show that observables are in a one-to-one correspondence with spectral (...)
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  13.  38
    On Bilinear Forms From the Point of View of Generalized Effect Algebras.Anatolij Dvurečenskij & Jiří Janda - 2013 - Foundations of Physics 43 (9):1136-1152.
    We study positive bilinear forms on a Hilbert space which are not necessarily bounded nor induced by some positive operator. We show when different families of bilinear forms can be described as a generalized effect algebra. In addition, we present families which are or are not monotone downwards (Dedekind upwards) σ-complete generalized effect algebras.
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  14.  28
    Connections Between BCK-Algebras and Difference Posetse.Anatolij Dvurečenskij & Hee Sik Kim - 1998 - Studia Logica 60 (3):421-439.
    We discuss the interrelations between BCK-algebras and posets with difference. Applications are given to bounded commutative BCK-algebras, difference posets, MV-algebras, quantum MV-algebras and orthoalgebras.
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  15.  14
    Bell-Type Inequalities and Orthomodular Lattices.Anatolij Dvurečenskij - 1999 - In Maria Luisa Dalla Chiara (ed.), Language, Quantum, Music. pp. 209--218.
  16.  13
    Smearing of Observables and Spectral Measures on Quantum Structures.Anatolij Dvurečenskij - 2013 - Foundations of Physics 43 (2):210-224.
    An observable on a quantum structure is any σ-homomorphism of quantum structures from the Borel σ-algebra of the real line into the quantum structure which is in our case a monotone σ-complete effect algebra with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean σ-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there corresponds a spectral measure.
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  17.  9
    Finitely Additive States and Completeness of Inner Product Spaces.Anatolij Dvurečenskij, Tibor Neubrunn & Sylvia Pulmannová - 1990 - Foundations of Physics 20 (9):1091-1102.
    For any unit vector in an inner product space S, we define a mapping on the system of all ⊥-closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. (...)
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  18.  2
    Lexicographic Pseudo MV-Algebras.Anatolij Dvurečenskij - 2015 - Journal of Applied Logic 13 (4):825-841.
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