Complete Effect Algebra Research Articles

Perfect Effect Algebras and Spectral Resolutions of Observables

We study perfect effect algebras, that is, effect algebras with the Riesz decomposition property where every element belongs either to its radical or to its co-radical. We define perfect effect algebras with principal radical and we show that the category of such effect algebras is categorically equivalent to the category of unital po-groups with interpolation. We introduce an observable on a $$\hbox {Rad}$$ -monotone $$\sigma $$ -complete perfect effect algebra with principal radical and we show that observables are in a one-to-one correspondence with spectral resolutions of observables.

Quantum Observables and Effect Algebras

We study observables on monotone $\sigma$-complete effect algebras. We find conditions when a spectral resolution implies existence of the corresponding observable. The set of sharp elements of a monotone $\sigma$-complete homogeneous effect algebra is a monotone $\sigma$-complete subalgebra. In addition, we study compatibility in orthoalgebras.

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

Remarks on effect-tribes

We show that an effect tribe of fuzzy sets ${\mathcal T}\subseteq [0,1]^X$ with the property that every $f\in {\mathcal T}$ is ${\mathcal B}_0({\mathcal T})$-measurable, where ${\mathcal B}_0({\mathcal T})$ is the family of subsets of $X$ whose characteristic functions are central elements in ${\mathcal T}$, is a tribe. Moreover, a monotone $\sigma$-complete effect algebra with RDP with a Loomis-Sikorski representation $(X, {\mathcal T},h)$, where the tribe ${\mathcal T}$ has the property that every $f\in {\mathcal T}$ is ${\mathcal B}_0({\mathcal T})$-measurable, is a $\sigma$-MV-algebra.

Top Element Problem and Macneille Completions of Generalized Effect Algebras

Effect algebras (EAs), introduced by D. J. Foulis and M. K. Bennett, as common generalizations of Boolean algebras, orthomodular lattices and MV-algebras, are nondistributive algebraic structures including unsharp elements. Their unbounded versions, called generalized effect algebras, are posets which may have or may have not an EA-MacNeille completion, or cannot be embedded into any complete effect algebra. We give a necessary and sufficient condition for a generalized effect algebra to have an EA-MacNeille completion. Some examples are provided.

Intervals in generalized effect algebras

A significant property of a generalized effect algebra is that its every interval with inherited partial sum is an effect algebra. We show that in some sense the converse is also true. More precisely, we prove that a set with zero element is a generalized effect algebra if and only if all its intervals are effect algebras. We investigate inheritance of some properties from intervals to generalized effect algebras, e.g., the Riesz decomposition property, compatibility of every pair of elements, dense embedding into a complete effect algebra, to be a sub-(generalized) effect algebra, to be lattice ordered and others. The response to the Open Problem from Rieă?anova and Zajac (2013) for generalized effect algebras and their sub-generalized effect algebras is given.

Smearing of Observables and Spectral Measures on Quantum Structures

An observable on a quantum structure is any $\sigma$-homomorphism of quantum structures from the Borel $\sigma$-algebra of the real line into the quantum structure which is in our case a monotone $\sigma$-complete effect algebras with the Riesz Decomposition Property. We show that every observable is a smearing of a sharp observable which takes values from a Boolean $\sigma$-subalgebra of the effect algebra, and we prove that for every element of the effect algebra there is its spectral measure.

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.

Representation of states on effect-tribes and effect algebras by integrals

We describe $\sigma$-additive states on effect-tribes by integrals. Effect-tribes are monotone $\sigma$-complete effect algebras of functions where operations are defined by points. Then we show that every state on an effect algebra is an integral through a Borel regular probability measure. Finally, we show that every $\sigma$-convex combination of extremal states on a monotone $\sigma$-complete effect algebra is a Jauch-Piron state.

Loomis–Sikorski theorem and Stone duality for effect algebras with internal state

Recently Flaminio and Montagna extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis–Sikorski Theorem for monotone σ -complete effect algebras with internal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order determining system of states is dual to the category of Bauer simplices Ω such that ∂ e Ω is an F-space.

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.

Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra $E$ is separable and modular then there exists a faithful state on $E$. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra $\widehat{E}$ and the compatiblity center of $E$ is not a Boolean algebra then there exists an $(o)$-continuous subadditive state on $E$.

State smearing theorems and the existence of states on some atomic lattice effect algebras

The existence of states and probabilities on effect algebras as logical structures when events may be non-compatible, unsharp, fuzzy or imprecise is still an open question. Only a few families of effect algebras possessing states are known. We are going to show some families of effect algebras, the existence of a pseudocomplementation on which implies the existence of states. Namely, those are Archimedean atomic lattice effect algebras, which are sharply dominating or s-compactly generated or extendable to complete lattice effect algebras.

Atoms and Dobrakov submeasures in effect algebras

The present paper deals with the study of the notion of Dobrakov submeasure m defined on an effect algebra L with values in [ 0 , ∞ ) . We have established the equivalence of the following three properties for a Dobrakov submeasure m defined on a σ -complete effect algebra L: (i) m is atomless, (ii) m has Saks property, (iii) m has Darboux property. We have also studied the concept of an atom of a function μ defined on an effect algebra L with values in [ 0 , ∞ ) and finally, we have proved for a monotone, m-continuous and semicontinuous function μ defined on a σ -complete effect algebra L that, μ is non-atomic if and only if μ is atomless.

The block structure of complete lattice ordered 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).

Loomis-Sikorski theorem for monotone σ-complete effect algebras

AbstractWe show that monotone σ -complete effect algebras under some conditions are σ - homomorphic images of effect-tribes (as monotone σ -complete effect algebras), which are nonempty systems of fuzzy sets closed under complements, sums of fuzzy sets less than 1, and containing all pointwise limits of nondecreasing fuzzy sets. Because effect-tribes are generalizations of Boolean σ -algebras of subsets, we present a generlization of the Loomis-Sikorski theorem for such effect algebras. We show that we can choose an effect-tribe to be a system of affin fuzzy sets. In addition, we present a new version of the Loomis-Sikorski theorem for σ-complete MV-algebras.

Loomis–Sikorski representation of monotone [formula omitted]-complete effect algebras

We show that monotone σ -complete effect algebras with the Riesz decomposition property are σ -homomorphic images of effect-tribes with the Riesz decomposition property, which are effect algebras of fuzzy sets closed under pointwise limits of nondecreasing fuzzy sets.

Embeddings of generalized effect algebras into complete effect algebras

Generalized effect algebras as posets are unbounded versions of effect algebras having bounded effect-algebraic extensions. We show that when the MacNeille completion MC(P) of a generalized effect algebra P cannot be organized into a complete effect algebra by extending the operation ⊕ onto MC(P) then still P may be densely embedded into a complete effect algebra. Namely, we show these facts for Archimedean GMV-effect algebras and block-finite prelattice generalized effect algebras. Moreover, we show that extendable commutative BCK-algebras directed upwards are equivalent to generalized MV-effect algebras.

Basic decomposition of elements and Jauch–Piron effect algebras

We show that every element of a complete atomic effect algebra E has a unique basic decomposition into a sum of a sharp element and unsharp multiples of isotropic atoms of E. Consequently, for such effect algebras we obtain “the Smearing Theorem for states” establishing that every order-continuous state existing on sharp elements of E can be extended to a state on E. For a σ -complete separable atomic effect algebra E we prove that E is a unital and Jauch–Piron effect algebra if and only if the set S ( E ) of all sharp elements of E is a unital Jauch–Piron orthomodular lattice and for finite E, S ( E ) is a Boolean algebra.

Lattice effect algebras with (o)-continuous faithful valuations

We prove that if there exists an order-continuous, faithful valuation ω on a lattice effect algebra E then E is modular, separable and order-continuous. It is also shown that such an effect algebra E can be supremum and infimum densely embedded into a complete effect algebra E ̂ which is also modular separable and order-continuous, since the valuation ω can be extended to a unique order-continuous faithful valuation ω ̂ on E ̂ .