Generalized Effect Algebras Research Articles

The exocenter and type decomposition of a generalized pseudoeffect algebra

We extend the notion of the exocenter of a generalized effect algebra (GEA) to a generalized pseudoeffect algebra (GPEA) and show that elements of the exocenter are in one-to-one correspondence with direct decompositions of the GPEA; thus the exocenter is a generalization of the center of a pseudoeffect algebra (PEA). The exocenter forms a boolean algebra and the central elements of the GPEA correspond to elements of a sublattice of the exocenter which forms a generalized boolean algebra. We extend the notion of central orthocompleteness to GPEA, prove that the exocenter of a centrally orthocomplete GPEA (COGPEA) is a complete boolean algebra and show that the sublattice corresponding to the center is a complete boolean subalgebra. We also show that in a COGPEA, every element admits an exocentral cover and that the family of all exocentral covers, the so-called exocentral cover system, has the properties of a hull system on a generalized effect algebra. We extend the notion of type determining (TD) sets, originally introduced for effect algebras and then extended to GEAs 14 D.J. Foulis, S. Pulmannova and E. Vincekova and PEAs, to GPEAs, and prove a type-decomposition theorem, analogous to the type decomposition of von Neumann algebras.

Hull determination and type decomposition for a generalized effect algebra

In this article, we show that certain generalized boolean subalgebras of the exocenter of a generalized effect algebra (GEA) determine hull systems on the GEA in a manner analogous to the determination of a hull mapping on an effect algebra (EA) by its set of invariant elements. We show that a hull system on a GEA E induces a hull mapping on each interval E[0, p] in E, and, using hull systems, we identify certain special elements of E (e.g., η-subcentral elements, η-monads, and η-dyads). We also extend the type-decomposition theory for EAs to GEAs.

On Realization of Generalized Effect Algebras

A well-known fact is that there is a finite orthomodular lattice with an order determining set of states which is not representable in the standard quantum logic, the lattice L(H) of all closed subspaces of a separable complex Hilbert space. We show that a generalized effect algebra is representable in the operator generalized effect algebra GD(H) of effects of a complex Hilbert space H iff it has an order determining set of generalized states. This extends the corresponding results for effect algebras of Riečanová and Zajac. Further, any operator generalized effect algebra GD(H) possesses an order determining set of generalized states.

Properties of Quasi-Hermitian Operators Inherited from Self-Adjoint Operators

We study a generalized effect algebra of unbounded linear operators in an infinite-dimensional complex Hilbert space. This algebra equipped with a certain kind of topology allows us to show that unbounded quasi-Hermitian operators can be expressed as a difference of two infinite sums of bounded quasi-Hermitian operators.

Topological Properties of Operator Generalized Effect Algebras

For any dense linear subspace D of a complex Hilbert space H we introduce and study a D-weak operator topology τD on the set G D ( H ) of all positive linear operators with domain D. For instance, we show that every A ∈ G D ( H ) is τD-limit of a nondecreasing sequence of bounded operators from G D ( H ) . Let τw denote the weak operator topology on the set of all bounded linear operators and let B + ( H ) denote the set of all positive bounded linear operators. For the corresponding relative topologies we show that τ D ∩ B + ( H ) ⊂ τ w ∩ B + ( H ) and the converse inclusion need not hold. Some topological properties of the operator generalized effect algebra ( G D ( H ) ; ⊕ , 0 ) and its intervals (all of which are effect algebras) are also studied.

The Exocenter of a Generalized Effect Algebra

Elements of the exocenter of a generalized effect algebra (GEA) correspond to decompositions of the GEA as a direct sum and thus the exocenter is a generalization to GEAs of the center of an effect algebra. The exocenter of a GEA is shown to be a boolean algebra, and the notion of a hull mapping for an effect algebra is generalized to a hull system for a GEA. We study Dedekind orthocompleteness of GEAs and extend to GEAs the notion of a centrally orthocomplete effect algebra.

Effect Algebras of Positive Linear Operators Densely Defined on Hilbert Spaces

We show that the set of all positive linear operators densely defined in an infinite-dimensional complex Hilbert space can be equipped with partial sum of operators making it a generalized effect algebra. This sum coincides with the usual sum of two operators whenever it exists. Moreover, blocks of this generalized effect algebra are proper sub-generalized effect algebras. All intervals in this generalized effect algebra become effect algebras which are Archimedean, convex, interval effect algebras, for which the set of vector states is order determining. Further, these interval operator effect algebras possess faithful states.

Generalized Effect Algebras of Bounded Positive Operators Defined on Hilbert Spaces

It is shown that compact positive operators, Hilbert–Schmidt positive operators and trace-class positive operators form sub-generalized effect algebras of the generalized effect algebra of positive bounded operators on Hilbert space (with the ⊕ operation being the usual operator sum) and they are generalized effect algebras in their own right. The intersections of these sets with the set of Hilbert space effects form three nontrivial sub-generalized effect algebras of the generalized effect algebra of Hilbert space effects. Also a more general version of these results is given.

Considerable Sets of Linear Operators in Hilbert Spaces as Operator Generalized Effect Algebras

We show that considerable sets of positive linear operators namely their extensions as closures, adjoints or Friedrichs positive self-adjoint extensions form operator (generalized) effect algebras. Moreover, in these cases the partial effect algebraic operation of two operators coincides with usual sum of operators in complex Hilbert spaces whenever it is defined. These sets include also unbounded operators which play important role of observables (e.g., momentum and position) in the mathematical formulation of quantum mechanics.

Extensions of Effect Algebra Operations

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.

More on PT-Symmetry in (Generalized) Effect Algebras and Partial Groups

We continue in the direction of our paper on PT-Symmetry in (Generalized) Effect Algebras and Partial Groups. Namely we extend our considerations to the setting of weakly ordered partial groups. In this setting, any operator weakly ordered partial group is a pasting of its partially ordered commutative subgroups of linear operators with a fixed dense domain over bounded operators. Moreover, applications of our approach for generalized effect algebras are mentioned.

$\mathcal{PT}$ -Symmetry in (Generalized) Effect Algebras

We show that an η+-pseudo-Hermitian operator for some metric operator η+ of a quantum system described by a Hilbert space \({\mathcal{H}}\) yields an isomorphism between the partially ordered commutative group of linear maps on \({\mathcal{H}}\) and the partially ordered commutative group of linear maps on \({\mathcal{H}}_{\rho_{+}}\). The same applies to the generalized effect algebras of positive operators and to the effect algebras of c-bounded positive operators on the respective Hilbert spaces \({\mathcal{H}}\) and \({\mathcal{H}}_{\rho_{+}}\). Hence, from the standpoint of (generalized) effect algebra theory both representations of our quantum system coincide.

Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.

Effectible residuated lattices and n-th roots

In this paper, conditions are given on residuated lattices and on Girard algebras in order for them to be generalized effect algebras and effect algebras, respectively. A new class of such “effectible” residuated lattices essentially enlarging MV-algebras is defined. Of particular interest are “pre-radicable” residuated lattices having all “ n-th roots” generalizing Höhle's square roots to an arbitrary order n. The existence of “partial” n-th roots on effectible residuated lattices and Girard algebras is related with the concept of divisible effect algebras.

Riesz ideals in generalized pseudo effect algebras and in their unitizations

We prove that there is an order isomorphism between the lattice of all normal Riesz ideals and the lattice of all Riesz congruences in upwards directed generalized pseudoeffect algebras (or GPEAs, for short). We give a sufficient and necessary condition under which a normal Riesz ideal I of a weak commutative generalized pseudoeffect algebra P is a normal Riesz ideal also in the unitization $$\widehat{P}$$of P. These results extend those obtained recently by Avalllone, Vitolo, Pulmannova and Vincekova for effect algebras. At the same time, we give the conditions under which the quotient of a generalized pseudoeffect algebra P is a generalized effect algebra and linearly ordered generalized pseudoeffect algebra.

Isomorphism theorems on generalized effect algebras based on atoms

A well-known fact is that every generalized effect algebra can be uniquely extended to an effect algebra in which it becomes a sub-generalized effect algebra and simultaneously a proper order ideal, the set-theoretic complement of which is its dual poset. We show that two non-isomorphic generalized effect algebras (even finite ones) may have isomorphic effect algebraic extensions. For Archimedean atomic lattice effect algebras we prove “Isomorphism theorem based on atoms”. As an application we obtain necessary and sufficient conditions for isomorphism of two prelattice Archimedean atomic generalized effect algebras with common (or isomorphic) effect algebraic extensions.

Effect algebraic extensions of generalized effect algebras and two-valued states

D-posets and effect algebras are very natural structures to be carriers of probabilities for events with fuzziness, uncertainty or unsharpness. In spite of that there are effect algebras admitting no states and probabilities. We give necessary and sufficient conditions for existence of two-valued states on Archimedean atomic effect algebras. Consequently a construction of states, probabilities from bounded orthogonally additive measures existing on some ideal sub-generalized effect algebras is shown.

Remarks on the order for quantum observables

Abstract Relations between generalized effect algebras and the sets of classical and quantum observables endowed with an ordering recently introduced in [GUDDER, S.: An order for quantum observables, Math. Slovaca 56 (2006), 573–589] are studied. In the classical case, a generalized OMP, while in the quantum case a weak generalized OMP is obtained. Existence of infima for arbitrary sets and suprema for above bounded sets in the quantum case is shown. Compatibility in the sense of Mackey is characterized.

Riesz ideals in generalized effect algebras and in their unitizations

We show that quotients of generalized effect algebras by Riesz ideals preserve some important special properties, e.g., homogeneity and hereditary Riesz decomposition properties; moreover, quotients of generalized orthoalgebras, weak generalized orthomodular posets, generalized orthomodular lattices and generalized MV-algebras with respect to Riesz ideals belong to the same class. We give a necessary and sufficient condition under which a Riesz ideal I of a generalized effect algebra P is a Riesz ideal also in the unitization E of P. We also study relations between Riesz ideals and central elements in GEAs and in their unitizations. In the last section, we demonstrate the notion of Riesz ideals by some illustrative examples.

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.