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Observables on $σ$ -MV algebras and $σ$ -lattice effect algebras

Anna Jenčová, Sylvia Pulmannová, Elena Vinceková (2011)

Kybernetika

Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered $σ$ -effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a $σ$ -MV algebra, and every observable is defined by a smearing of a sharp...

On $0 - 1$ measure for projectors

Václav Alda (1980)

Aplikace matematiky

An example of a finite set of projectors in $E_{3}$ is exhibited for which no 0-1 measure exists.

On a certain mapping on the set with orthogonality

Jan Havrda (1989)

Časopis pro pěstování matematiky

On a characterization of probability measures on Boolean algebras and some orthomodular lattices

Helmut Länger, Maciej Mączyński (1995)

Mathematica Slovaca

On a sum of observables in a logic

Anatolij Dvurečenskij (1980)

Mathematica Slovaca

On a summability of observables in generalized measurable spaces

Nadežda Chrapčiaková (1987)

Mathematica Slovaca

On centers and state spaces of logics

Pták, Pavel (1984)

Proceedings of the 11th Winter School on Abstract Analysis

On central atoms of Archimedean atomic lattice effect algebras

Martin Kalina (2010)

Kybernetika

If element $z$ of a lattice effect algebra $(E, \oplus, 0, 1)$ is central, then the interval $[0, z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$ . It is a known fact that if the set $C (E)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C (E)$ in $E$ equals to the top element of $E$ , then $E$ is isomorphic to a direct product of irreducible effect algebras ([16]). In [10] Paseka and Riečanová published as open problem whether $C (E)$ is a bifull sublattice...

On complete-cocomplete subspaces of an inner product space

David Buhagiar, Emmanuel Chetcuti (2005)

Applications of Mathematics

In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space $S$ is complete if and only if there exists a $σ$ -additive state on $C (S)$ , the orthomodular poset of complete-cocomplete subspaces of $S$ . We then consider the problem of whether every state on $E (S)$ , the class of splitting subspaces of $S$ , can be extended to a Hilbertian state on $E (\bar{S})$ ; we show that for the dense hyperplane $S$ (of a separable Hilbert space) constructed by P. Pták and...

On extension properties for observables

Anatolij Dvurečenskij (1981)

Mathematica Slovaca

On generating and concreteness in quantum logics

Vladimír Rogalewicz (1991)

Mathematica Slovaca

On isomorphisms of inner product spaces

David Buhagiar, Emmanuel Chetcuti (2004)

Mathematica Slovaca

On joint distribution in quantum logics. I. Compatible observables

Anatolij Dvurečenskij (1987)

Aplikace matematiky

The notion of a joint distribution in $σ$ -finite measures of observables of a quantum logic defined on some system of $σ$ -independent Boolean sub- $σ$ -algebras of a Boolean $σ$ -algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.

On joint distribution in quantum logics. II. Noncompatible observables

Anatolij Dvurečenskij (1987)

Aplikace matematiky

This paper i a continuation of the first part under the same title. The author studies a joint distribution in $σ$ -finite measures for noncompatible observables of a quantum logic defined on some system of $σ$ -independent Boolean sub- $σ$ -algebras of a Boolean $σ$ -algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system...

On $m$ -joint distribution

Anatolij Dvurečenskij (1981)

Mathematica Slovaca

On mean value in $F$ -quantum spaces

Beloslav Riečan (1990)

Aplikace matematiky

The paper deals with a new mathematical model for quantum mechanics based on the fuzzy set theory [1]. The indefinite integral of observables is defined and some basic properties of the integral are examined.

On quantic conuclei in orthomodular lattices.

Román, Leopoldo, Zuazua, Rita E. (1996)

Theory and Applications of Categories [electronic only]

On representations of logics

Sylvia Pulmannová (1983)

Mathematica Slovaca

On some contributions to quantum structures by fuzzy sets

Beloslav Riečan (2007)

Kybernetika

It is well known that the fuzzy sets theory can be successfully used in quantum models ([5, 26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations – multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.

On some properties of submeasures on MV-algebras

Mária Jurečková, Ferdinand Chovanec (2004)

Mathematica Slovaca

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