scholarly journals The block structure of complete lattice ordered effect algebras

2007 ◽  
Vol 83 (2) ◽  
pp. 181-216 ◽  
Author(s):  
Gejza Jenča
Keyword(s):  
Complete Lattice ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Block Structure ◽  
Effect Algebras

AbstractWe prove that every for every complete lattice-ordered effect algebra E there exists an orthomodular lattice O(E) and a surjective full morphism øE: O(E) → E which preserves blocks in both directions: the (pre)imageofa block is always a block. Moreover, there is a 0, 1-lattice embedding : E → O(E).

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.

On realization of effect algebras

2016 ◽  
Vol 66 (2) ◽  
Author(s):  
Josef Niederle ◽  
Jan Paseka
Keyword(s):  
Quantum Logic ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Simplex Algorithm ◽  
Effect Algebras ◽  
Standard Quantum ◽  
Generalized Effect Algebra ◽  
Determining Set

AbstractA well known fact is that there is a finite orthomodular lattice with an order determining set of states which is not order embeddable into the standard quantum logic, the latticeWe show that a finite generalized effect algebra is order embeddable into the standard effect algebraAs an application we obtain an algorithm, which is based on the simplex algorithm, deciding whether such an order embedding exists and, if the answer is positive, constructing it.

scholarly journals Separating points of measures on effect algebras

2007 ◽  
Vol 57 (2) ◽  
Author(s):  
Anna Avallone
Keyword(s):  
Complete Lattice ◽  
Effect Algebra ◽  
Countable Family ◽  
Effect Algebras

AbstractWe prove the existence of separating points for every countable family of nonatomic σ-additive modular measures on a σ-complete lattice ordered effect algebra.

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 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.

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.

scholarly journals Roughness in Lattice Ordered Effect Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Xiao Long Xin ◽  
Xiu Juan Hua ◽  
Xi Zhu
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Algebraic Systems ◽  
Riesz Ideal ◽  
Congruence Classes

Many authors have studied roughness on various algebraic systems. In this paper, we consider a lattice ordered effect algebra and discuss its roughness in this context. Moreover, we introduce the notions of the interior and the closure of a subset and give some of their properties in effect algebras. Finally, we use a Riesz ideal induced congruence and define a functione(a,b)in a lattice ordered effect algebraEand build a relationship between it and congruence classes. Then we study some properties about approximation of lattice ordered effect algebras.

scholarly journals Annihilators in JB-algebras

1990 ◽  
Vol 108 (2) ◽  
pp. 317-323 ◽  
Author(s):  
M. Battaglia
Keyword(s):  
Complete Lattice ◽  
Orthomodular Lattice ◽  
Complete Orthomodular Lattice

AbstractOrthogonality is defined for all elements in a JB-algebra and Topping's results on annihilators in JW-algebras are generalized to the context of JB- and JBW-algebras. A pair (a, b) of elements in a JB-algebra A is said to be orthogonal provided that a2 ∘ b equals zero. It is shown that this relation is symmetric. The annihilator S⊥ of a subset S of A is defined to be the set of elements a in A such that, for all elements s in S, the pair (s, a) is orthogonal. It is shown that the annihilators are closed quadratic ideals and, if A is a JBW-algebra, a subset I of A is a w*-closed quadratic ideal if and only if I coincides with its biannihilator I⊥⊥. Moreover, in a JBW-algebra A the formation of the annihilator of a w*-closed quadratic ideal is an orthocomplementation on the complete lattice of w*-closed quadratic ideals which makes it into a complete orthomodular lattice. Further results establish a connection between ideals, central idempotents and annihilators in JBW-algebras.

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