scholarly journals Effect algebras, witness pairs and observables

10.29007/71gb ◽  
2018 ◽  
Author(s):  
Gejza Jenča
Keyword(s):  
Boolean Algebra ◽  
Open Problem ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Orthomodular Posets ◽  
Interval Effect ◽  
Bounded Posets

The category of effect algebras is the Eilenberg-Moore category for themonad arising from the free-forgetful adjunction between categories of bounded posetsand orthomodular posets.In the category of effect algebras, an observable is a morphism whose domainis a Boolean algebra. The characterization of subsets of ranges of observables isan open problem.For an interval effect algebra E, a witness pair for a subset of S is an objectliving within E that "witnesses existence" of an observable whose rangeincludes S. We prove that there is an adjunction between the poset of allwitness pairs of E and the category of all partially inverted E-valuedobservables.

scholarly journals A Note of Filters in Effect Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Biao Long Meng ◽  
Xiao Long Xin
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Lattice Filter ◽  
Mv Algebra ◽  
Mv Algebras

We investigate relations of the two classes of filters in effect algebras (resp., MV-algebras). We prove that a lattice filter in a lattice ordered effect algebra (resp., MV-algebra) does not need to be an effect algebra filter (resp., MV-filter). In general, in MV-algebras, every MV-filter is also a lattice filter. Every lattice filter in a lattice ordered effect algebra is an effect algebra filter if and only if is an orthomodular lattice. Every lattice filter in an MV-algebra is an MV-filter if and only if is a Boolean algebra.

The generalized orthomodularity property: configurations and pastings

2020 ◽  
Vol 30 (5) ◽  
pp. 991-1022
Author(s):  
Ivan Chajda ◽  
Davide Fazio ◽  
Antonio Ledda
Keyword(s):  
Theoretical Analysis ◽  
Effect Algebras ◽  
Orthomodular Posets ◽  
Theoretical Characterization

Abstract In this paper, we consider a generalization of the notion of orthomodularity for posets to the concept of the generalized orthomodularity property (GO-property) by considering the $LU$-operators. This seemingly mild generalization of orthomodular posets and its order theoretical analysis yield rather strong application to effect algebras and orthomodular structures. Also, for several classes of orthoalgebras, the GO-property yields a completely order-theoretical characterization of the coherence law, and, in turn, of proper orthoalgebras.

The pasting constructions for effect algebras

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Yongjan Xie ◽  
Yongming Li ◽  
Aili Yang
Keyword(s):  
Boolean Algebra ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Finite Lattice ◽  
Effect Algebras ◽  
The Family

AbstractThe aim of this paper is to present several techniques of constructing a lattice-ordered effect algebra from a given family of lattice-ordered effect algebras, and to study the structure of finite lattice-ordered effect algebras. Firstly, we prove that any finite MV-effect algebra can be obtained by substituting the atoms of some Boolean algebra by linear MV-effect algebras. Then some conditions which can guarantee that the pasting of a family of effect algebras is an effect algebra are provided. At last, we prove that any finite lattice-ordered effect algebra E without atoms of type 2 can be obtained by substituting the atoms of some orthomodular lattice by linear MV-effect algebras. Furthermore, we give a way how to paste a lattice-ordered effect algebra from the family of MV-effect algebras.

scholarly journals Extensions of Effect Algebra Operations

10.14311/1410 ◽  
2011 ◽  
Vol 51 (4) ◽  
Author(s):  
Z. Riečanová ◽  
M. Zajac
Keyword(s):  
Effect Algebra ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Effect Algebras ◽  
Infinite Dimensional ◽  
Interval Effect ◽  
Generalized Effect Algebra ◽  
Algebraic Operations ◽  
Densely Defined ◽  
Generalized Effect Algebras

We study the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space. We equip this set with various effect algebraic operations making it a generalized effect algebra. Further, sub-generalized effect algebras and interval effect algebras with respect of these operations are investigated.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

scholarly journals Module Structure on Effect Algebras

2020 ◽  
Author(s):  
Simin Saidi Goraghani ◽  
Rajab Ali Borzooei
Keyword(s):  
Effect Algebra ◽  
Module Structure ◽  
Effect Algebras ◽  
Product Effect

 In this paper, by considering the notions of effect algebra and product effect algebra, we define the concept of effect module. Then we investigate some properties of effect modules, and we present some examples on them. Finally, we introduce some interesting topologies on effect modules.

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.

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.

An algebraic characterization of power set in countable standard models of ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 167-170
Author(s):  
George Metakides ◽  
J. M. Plotkin
Keyword(s):  
Standard Model ◽  
Boolean Algebra ◽  
Classical Result ◽  
Algebraic Characterization ◽  
Categorical Theory ◽  
Power Set ◽  
Standard Models ◽  
Image Position ◽  
Atomic Boolean Algebra

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

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