scholarly journals Extensions of ordering sets of states from effect algebras onto their MacNeille completions

10.29007/lkdv ◽  
2018 ◽  
Author(s):  
Jiří Janda ◽  
Zdenka Riečanová
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Macneille Completion ◽  
Atomic Lattice ◽  
Space Effect ◽  
Lattice Effect

In [Riečanová Z, Zajac M.: Hilbert Space Effect-Representations of Effect Algebras] it was shown that an effect algebra E with an ordering set M of states can by embedded into a Hilbert space effect algebra E(l<sub>2</sub>(M)). We consider the problem when its effect algebraic MacNeille completion Ê can be also embedded into the same Hilbert space effect algebra E(l<sub>2</sub>(M)). That is when the ordering set M of states on E can be be extended to an ordering set of states on Ê. We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.

scholarly journals Coexistency on Hilbert Space Effect Algebras and a Characterisation of Its Symmetry Transformations

2020 ◽  
Vol 379 (3) ◽  
pp. 1077-1112 ◽  
Author(s):  
György Pál Gehér ◽  
Peter Šemrl
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Mathematical Structure ◽  
Quantum Measurements ◽  
The Other ◽  
Effect Algebras ◽  
Natural Question ◽  
Fuzzy Event ◽  
Space Effect ◽  
Symmetry Transformations

Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.

scholarly journals Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect Algebras

10.14311/1398 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
M. Kalina
Keyword(s):  
Effect Algebra ◽  
Subdirect Product ◽  
Sufficient Conditions ◽  
Effect Algebras ◽  
Lattice Effect Algebra ◽  
Orthomodular Lattices ◽  
Atomic Lattice ◽  
Lattice Effect ◽  
Archimedean Lattice ◽  
Mv Algebras

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras. This means that within lattice effect algebras it is possible to model such effects as unsharpness (fuzziness) and/or non-compatibility. The main problem is the existence of a state. There are lattice effect algebras with no state. For this reason we need some conditions that simplify checking the existence of a state. If we know that the center C(E) of an atomic Archimedean lattice effect algebra E (which is again atomic) is a bifull sublattice of E, then we are able to represent E as a subdirect product of lattice effect algebras Ei where the top element of each one of Ei is an atom of C(E). In this case it is enough if we find a state at least in one of Ei and we are able to extend this state to the whole lattice effect algebra E. In [8] an atomic lattice effect algebra E (in fact, an atomic orthomodular lattice) with atomic center C(E) was constructed, where C(E) is not a bifull sublattice of E. In this paper we show that for atomic lattice effect algebras E (atomic orthomodular lattices) neither completeness (and atomicity) of C(E) nor σ-completeness of E are sufficient conditions for C(E) to be a bifull sublattice of E.

scholarly journals Sharply Orthocomplete Effect Algebras

10.14311/1267 ◽  
2010 ◽  
Vol 50 (5) ◽  
Author(s):  
M. Kalina ◽  
J. Paseka ◽  
Z. Riečanová
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Lattice Effect Algebra ◽  
Atomic Lattice ◽  
Complete Lattices ◽  
Lattice Effect

Special types of effect algebras E called sharply dominating and S-dominating were introduced by S. Gudder in [7, 8]. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of E. Namely we prove that in every sharply orthocomplete S-dominating effect algebra E the set of sharp elements and the center of E are complete lattices bifull in E. If an Archimedean atomic lattice effect algebra E is sharply orthocomplete then it is complete.

scholarly journals Generalized Jordan Semitriple Maps on Hilbert Space Effect Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Qing Yuan ◽  
Kan He
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Space Effect ◽  
Antiunitary Operator

Letℰ(H)be the Hilbert space effect algebra on a Hilbert spaceHwithdim⁡H≥3,α,βtwo positive numbers with2α+β≠1andΦ:ℰ(H)→ℰ(H)a bijective map. We show that ifΦ(AαBβAα)=Φ(A)αΦ(B)βΦ(A)αholds for allA,B∈ℰ(H), then there exists a unitary or an antiunitary operatorUonHsuch thatΦ(A)=UAU*for everyA∈ℰ(H).

Properties of implication in effect algebras

2021 ◽  
Vol 71 (3) ◽  
pp. 523-534
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
State Space ◽  
Effect Algebra ◽  
Modus Ponens ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Deductive Systems ◽  
Algebraic Description ◽  
Residuated Poset

Abstract Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

Dynamic effect algebras

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Ivan Chajda ◽  
Miroslav Kolařík
Keyword(s):  
Quantum Mechanics ◽  
Effect Algebra ◽  
Dynamic Effect ◽  
Effect Algebras ◽  
Lattice Effect Algebra ◽  
Tense Operators ◽  
Lattice Effect

AbstractWe introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.

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