Anatolij Dvurečenskij

New trends in quantum structures

Preface. Introduction. 1. D-posets and Effect Algebras. 2. MV-algebras and QMV-algebras. 3. Quotients of Partial Abelian Monoids. 4. Tensor Product of D-Posets and Effect Algebras. 5. BCK-algebras. 6. BCK-algebras in Applications. 7. Loomis-Sikorski Theorems for MV-algebras and BCK-algebras. Bibliography. Index of Symbols. Index. List of Figures and Tables.

Pseudo MV-algebras are intervals in ℓ-groups

We show that any pseudo MV-algebra is isomorphic with an interval Γ(G, u), where G is an l-group not necessarily Abelian with a strong unit u. In addition, we prove that the category of unital l-groups is categorically equivalent with the category of pseudo MV-algebras. Since pseudo MV-algebras are a non-commutative generalization of MV-algebras, our assertions generalize a famous result of Mundici for a representation of MV-algebras by Abelian unital l-groups. Our methods are completely different from those of Mundici. In addition, we show that any Archimedean pseudo MV-algebra is an MV-algebra.

States on Pseudo MV-Algebras

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.

Pseudoeffect Algebras. I. Basic Properties

As a noncommutative generalization of effect algebras, we introduce pseudoeffect algebras and list some of their basic properties. For the purpose of a structure theory, we further define several kinds of Riesz-like properties for pseudoeffect algebras and show how they are interrelated.

Pseudoeffect Algebras. II. Group Representations

This paper is the continuation of the previous paper by Dvurečenskij and Vetterlein (2001), Int. J. Theor. Phys. 40(3). We show that any pseudoeffect algebra fulfilling a certain property of Riesz type is representable by a unit interval of some (not necessarily Abelian) partially ordered group. The relation of pseudoeffect to pseudo-MV algebras is made clear, and the &ell-group representation theorem for the latter structure is re-proved.

State-morphism MV-algebras

Abstract 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-morphism) 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.

Difference posets, effects, and quantum measurements

Difference posets as generalizations of quantum logics, orthoalgebras, and effects are studied. Observables and measures generalizing normalized POV-measures and generalized measures on sets of effects are introduced. Characterization of orthomodularity of subsets of a difference poset in terms of triangle closedness and regularity of these subsets enables us to characterize observables with a Boolean range. Boolean powers of difference posets are investigated; they have similar properties to that of tensor products, and their connection with quantum measurements is studied.

Every Linear Pseudo BL-Algebra Admits a State

We show that every linear pseudo BL-algebra, hence every representable one, admits a state and is good. This solves positively the problem on the existence of states raised in Dvurečenskij and Rachůnek (Probabilistic averaging in bounded communitative residuated ℓ-monoids, 2006), and gives a partial answer to the problem on good pseudo BL-algebras from [Di Nola, Georgescu and Iorgulescu (Multiple Val Logic 8:715–750, 2002) Problem 3.21]. Moreover, we present that every saturated linear pseudo BL-algebra can be expressed as an ordinal sum of Hájek’s type of irreducible pseudo linear pseudo BL-algebras.

Algebras in the Positive Cone of po-Groups

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text.All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete.Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.

Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras

Pseudo-effect algebras are partial algebras ( E ; +, 0, 1) with a partially defined addition + which is not necessary commutative and with two complements, left and right ones. We define central elements of a pseudo-effect algebra and the centre, which in the case of MV-algebras coincides with the set of Boolean elements and in the case of effect algebras with the Riesz decomposition property central elements are only characteristic elements. If E satisfies general comparability, then E is a pseudo MV-algebra. Finally, we apply central elements to obtain a variation of the Cantor-Bernstein theorem for pseudo-effect algebras.