scholarly journals Closure systems over effect algebras

2021 ◽  
Vol 34 (02) ◽  
pp. 733-743
Author(s):  
Mahdi Ronasi ◽  
Esfandiar Eslami
Keyword(s):  
Effect Algebra ◽  
Effect Algebras ◽  
Arbitrary Subset ◽  
Closure System ◽  
Closure Systems

The present paper is an attempt to introduce the closure systems over effect algebras. At first, we will define closure systems over effect algebras, and for arbitrary set $ U $ and arbitrary subset S of all functions from U to an effect algebra L we will obtain the closure system containing S. Then, we will define the base of this closure system, and for arbitrary subset S of all functions from U to an effect algebra L we will obtain the base of this closure system.

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 Closures on partial partitions from closures on sets

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Christian Ronse
Keyword(s):  
Closure Operator ◽  
Closure System ◽  
Partial Partition ◽  
Closure Systems ◽  
Partial Partitions

AbstractJordens and Sturm investigated the link between closure systems on sets and closure systems on partitions. We extend that study to the wider framework of partial partitions, and highlight better the relation between these two families of closure systems. Then we consider the construction of a closure operator on partial partitions by the iterated application a set operator to the blocks of a partial partition; the resulting closure system fits into our framework.

scholarly journals Effect algebras with state operator

10.29007/jbdq ◽  
2018 ◽  
Author(s):  
Silvia Pulmannova
Keyword(s):  
Effect Algebra ◽  
Operator Algebras ◽  
Internal State ◽  
Effect Algebras ◽  
State Operator ◽  
Conditional Expectations ◽  
Definition Of ◽  
Mv Algebras

A state operator on effect algebras is introduced as an additive, idempotent and unital mapping from the effect algebra into itself. The definition is inspired by the definition of an internal state on MV-algebras, recently introduced by Flaminio and Montagna. We study state operators on convex effect algebras, and show their relations with conditional expectations on operator algebras.

Effect algebras with a full set of states

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Ivan Chajda
Keyword(s):  
Temporal Logic ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Tense Operators ◽  
Numerical Algebra ◽  
Full Set Of States

AbstractIt is shown that every effect algebra with a full set of states can be represented as a so-called numerical algebra introduced in the paper. For numerical algebras there are introduced tense operators which indicate dynamical changes of quantum events depending on variability of states. These operators enable to recognize an effect algebra with a full set of states as a temporal logic where events are quantified by these tense operators. The problem of representation of tense operators on a given numerical algebra is solved.

STATES, UNIFORMITIES AND METRICS ON LATTICE EFFECT ALGEBRAS

2002 ◽  
Vol 10 (supp01) ◽  
pp. 125-133 ◽  
Author(s):  
ZDENKA RIEČANOVÁ
Keyword(s):  
Effect Algebra ◽  
Order Topology ◽  
Effect Algebras ◽  
Lattice Effect Algebra ◽  
Uniform Topology ◽  
Lattice Effect

We show that every state ω on a lattice effect algebra E induces a uniform topology on E. If ω is subadditive this topology coincides with pseudometric topology induced by ω. Further, we show relations between the interval and order topology on E and topologies induced by states.

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.

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