Lattice Effect Algebras Research Articles

Olson Order of Quantum Observables

Using ideas of Olson \cite{Ols} who showed that the system of effect operators of a Hilbert space can be ordered by the so-called spectral order such that the system of effect operators is a complete lattice. Using his ideas, we introduce a partial order, called the Olson order, on the set of bounded observables of a complete lattice effect algebra. We show that the set of bounded observables is a Dedekind complete lattice.0

Weakly algebraic ideal topology of effect algebras

In this paper, we show that every weakly algebraic ideal of an effect algebra E induces a uniform topology (weakly algebraic ideal topology, for short) with which E is a first-countable, zero-dimensional, disconnected, locally compact and completely regular topological space, and the operation ⊕ of effect algebras is continuous with respect to these topologies. In addition, we prove that the operation ⊝ of effect algebras and the operations ∧ and ∨ of lattice effect algebras are continuous with respect to the weakly algebraic ideal topology generated by a Riesz ideal.

How to Produce S-Tense Operators on Lattice Effect Algebras

Tense operators in effect algebras play a key role for the representation of the dynamics of formally described physical systems. For this, it is important to know how to construct them on a given effect algebra $$ E$$ and how to compute all possible pairs of tense operators on $$ E$$ . However, we firstly need to derive a time frame which enables these constructions and computations. Hence, we usually apply a suitable set of states of the effect algebra $$ E$$ in question. To approximate physical reality in quantum mechanics, we use only the so-called Jauch–Piron states on $$ E$$ in our paper. To realize our constructions, we are restricted on lattice effect algebras only.

On States and State Operators on Certain Basic Algebras

The paper is based on the authors’ talks given at the 11th IQSA Meeting in Cagliari; it deals with states and state operators (internal states) on basic algebras which are a generalization of MV-algebras and orthomodular lattices, also including lattice effect algebras. On the one hand, the paper is a survey of some previous results on states and state operators on commutative basic algebras, and on the other one, an extension of these results to the much larger class of basic algebras where the addition ⊕ distributes over the lattice meet ∧.

Finitely generated varieties of distributive effect algebras

Lattice-ordered effect algebras generalize both MV-algebras and orthomodular lattices. In this paper, finitely generated varieties of distributive lattice effect algebras are axiomatized, and for any positive integer n, the free n-generator algebras in these varieties are described.

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions

In (Riecanova and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set $\mathcal{M}$ of states can by embedded into a Hilbert space effect algebra $\mathcal{E}(l_{2}(\mathcal{M}))$ . We consider the problem when its effect algebraic MacNeille completion $\hat{E}$ can be also embedded into the same Hilbert space effect algebra $\mathcal {E}(l_{2}(\mathcal{M}))$ . That is when the ordering set $\mathcal{M}$ of states on E can be extended to an ordering set of states on $\hat{E}$ . We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.

Almost Orthogonality and Hausdorff Interval Topologies of de Morgan Lattices and Lattice Effect Algebras

The topologies on ordered structures have been intensively studied by mathematicians and computer scientists. Various types of topologies may be introduced, depending on the nature of the ordered sets considered. Our purpose here is to study the interval topology tau_i , the order topology tau_o and the topology tau_phi induced by a canonical intrinsic uniformity generated by a certain family of pseudometrics on de Morgan lattices. This uniformity and topology may be regarded as a “two-sided symmetrization” of a similar intrinsic uniformity introduced by Erne and Palko for an order-theoretical construction of certain uniform completions. We prove that on a de Morgan lattice L with a join-dense set U the interval topology tau_i is Hausdorff and L is compactly generated by the elements of U if and only if L is U-almost orthogonal if and only if any element of U is hypercompact.

MacNeille completion of centers and centers of MacNeille completions of lattice effect algebras: Generic scheme behind

Abstract We present an example of a natural class of atomic Archimedean sharply dominating lattice effect algebras with non-bifull and atomic centers. Further, we extend and clarify known results about MacNeille completion of centers and centers of MacNeille completions of lattice effect algebras.

Logical Connectives on Lattice Effect Algebras

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hajek's BL-algebras, and the basic algebras of Chajda, Halas, and Kuhr.

Homogeneous orthocomplete effect algebras are covered by MV-algebras

The aim of our paper is twofold. First, we thoroughly study the sets of meager and hypermeager elements. Second, we study a common generalization of orthocomplete and lattice effect algebras. We show that every block of an Archimedean homogeneous effect algebra satisfying this generalization is lattice ordered. Hence such effect algebras can be covered by ranges of observables. As a corollary, this yields that every block of a homogeneous orthocomplete effect algebra is lattice ordered. Therefore finite homogeneous effect algebras are covered by MV-algebras.

Congruences generated by ideals of the compatibility center of lattice effect algebras

We prove that for every lattice effect algebra, the system of all congruences generated by the prime ideals of the compatibility center separates the elements. This is a common generalization of Chang's representation theorem from 1959 and a result of Graves and Selesnick (Colloq Math 27:21---30, 1973).

A dynamic Effect Algebras with Dual Operation

Tense operators for MV-algebras were introduced by Diaconescu and Georgescu. Based on their denition Chajda and Kolařik presented the denition of tense operators for lattice effect algebras. Chajda and Paseka tackled the problem of axiomatizing tense operators on an effect algebra by introducing the notion of a partial dynamic effect algebra. They also gave representation theorems for dynamic effect algebras. We continue to extend their work for partial S-dynamic effect algebras i.e. in the case when tense operators satisfy required conditions also for the dual effect algebraic operation . We show that whenever tense operators are total our stronger notion coincides with their denition. We give also a representation theorem for partial S-dynamic effect algebras and its version for strict dynamic effect algebras.

Dynamic effect algebras and their representations

For lattice effect algebras, the so-called tense operators were already introduced by Chajda and Kolařik. 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 in every effect algebra can be introduced tense operators which, for non-complete lattice effect algebras, can be only partial mappings. An effect algebra equipped with tense operators reflects changes of quantum events from past to future. A crucial problem concerning tense operators is their representation. Having an effect algebra with tense operators, we can ask if there exists a frame such that each of these operators can be obtained by our construction. We solve this problem for (strict) dynamic effect algebras having a full set of homorphisms into a complete lattice effect algebra.

Dynamic effect algebras

Abstract We 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.

Pre-ideals of Basic Algebras

Basic algebras are a generalization of MV-algebras, also including orthomodular lattices and lattice effect algebras. A pre-ideal of a basic algebra is a non-empty subset that is closed under the addition ⊕ and downwards closed with respect to the underlying order. In this paper, we study the pre-ideal lattices of algebras in a particular subclass of basic algebras which are closer to MV-algebras than basic algebras in general. We also prove that finite members of this subclass are exactly finite MV-algebras.

More about sharp and meager elements in Archimedean atomic lattice effect algebras

The aim of our paper is twofold. First, we thoroughly study the set of meager elements $$\hbox{M}(E),$$ the center $$\hbox{C}(E)$$ and the compatibility center $$\hbox{B}(E)$$ in the setting of atomic Archimedean lattice effect algebras $$E.$$ The main result is that in this case the center $$\hbox{C}(E)$$ is bifull (atomic) iff the compatibility center $$\hbox{B}(E)$$ is bifull (atomic) whenever $$E$$ is sharply dominating. As a by-product, we give a new description of the smallest sharp element over $$x\in E$$ via the basic decomposition of $$x.$$ Second, we prove the Triple Representation Theorem for sharply dominating atomic Archimedean lattice effect algebras.

Lattice pseudoeffect algebras as double residuated structures

Pseudoeffect algebras are partial algebraic structures which are non-commutative generalizations of effect algebras. The main result of the paper is a characterization of lattice pseudoeffect algebras in terms of so-called pseudo Sasaki algebras. In contrast to pseudoeffect algebras, pseudo Sasaki algebras are total algebras. They are obtained as a generalization of Sasaki algebras, which in turn characterize lattice effect algebras. Moreover, it is shown that lattice pseudoeffect algebras are a special case of double CI-posets, which are algebraic structures with two pairs of residuated operations, and which can be considered as generalizations of residuated posets. For instance, a lattice ordered pseudoeffect algebra, regarded as a double CI-poset, becomes a residuated poset if and only if it is a pseudo MV-algebra. It is also shown that an arbitrary pseudoeffect algebra can be described as a special case of conditional double CI-poset, in which case the two pairs of residuated operations are only partially defined.

Note on Generalizations of Orthocomplete and Lattice Effect Algebras

Note on Generalizations of Orthocomplete and Lattice Effect Algebras

Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect 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.

Order Topology and Frink Ideal Topology of Effect Algebras

In this paper, the following results are proved: (1) If E is a complete atomic lattice effect algebra, then E is (o)-continuous iff E is order-topological iff E is totally order-disconnected iff E is algebraic. (2) If E is a complete atomic distributive lattice effect algebra, then its Frink ideal topology τ id is Hausdorff topology and τ id is finer than its order topology τ o , and τ id =τ o iff 1 is finite iff every element of E is finite iff τ id and τ o are both discrete topologies. (3) If E is a complete (o)-continuous lattice effect algebra and the operation ⊕ is order topology τ o continuous, then its order topology τ o is Hausdorff topology. (4) If E is a (o)-continuous complete atomic lattice effect algebra, then ⊕ is order topology continuous.