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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 central atoms of Archimedean atomic lattice effect algebras

Martin Kalina (2010)

Kybernetika

If element z of a lattice effect algebra ( E , ⊕ , 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 ⊕ . 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 interval homogeneous orthomodular lattices

Anna de Simone, Mirko Navara, Pavel Pták (2001)

Commentationes Mathematicae Universitatis Carolinae

An orthomodular lattice L is said to be interval homogeneous (resp. centrally interval homogeneous) if it is σ -complete and satisfies the following property: Whenever L is isomorphic to an interval, [ a , b ] , in L then L is isomorphic to each interval [ c , d ] with c ≤ a and d ≥ b (resp. the same condition as above only under the assumption that all elements a , b , c , d are central in L ). Let us denote by Inthom (resp. Inthom c ) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous...

On Kalmbach measurability

A. B. d' Andrea, P. de Lucia, John David Maitland Wright (1994)

Applications of Mathematics

In this note we show that, for an arbitrary orthomodular lattice L , when μ is a faithful, finite-valued outer measure on L , then the Kalmbach measurable elements of L form a Boolean subalgebra of the centre of L .

On systems of congruences on principal filters of orthomodular implication algebras

Radomír Halaš, Luboš Plojhar (2007)

Mathematica Bohemica

Orthomodular implication algebras (with or without compatibility condition) are a natural generalization of Abbott’s implication algebras, an implication reduct of the classical propositional logic. In the paper deductive systems (= congruence kernels) of such algebras are described by means of their restrictions to principal filters having the structure of orthomodular lattices.

On the concreteness of quantum logics

Pavel Pták, John David Maitland Wright (1985)

Aplikace matematiky

It is shown that for any quantum logic L one can find a concrete logic K and a surjective homomorphism f from K onto L such that f maps the centre of K onto the centre of L . Moreover, one can ensure that each finite set of compatible elements in L is the image of a compatible subset of K . This result is “best possible” - let a logic L be the homomorphic image of a concrete logic under a homomorphism such that, if F is a finite subset of the pre-image of a compatible subset of L , then F is compatible....

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