Orthocomplete Effect Algebra Research Articles

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.

Quotients of dimension effect algebras

We show that the quotient of a dimension effect algebra by its dimension equivalence relation is a unital bounded lattice-ordered positive partial abelian monoid that satisfies a version of the Riesz decomposition property. For a dimension effect algebra of finite type, the quotient is a centrally orthocomplete Stone–Heyting MV-effect algebra; moreover, an orthocomplete effect algebra in which equality is a dimension equivalence relation is the same thing as a complete Stone–Heyting MV-effect algebra.

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.

TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

AbstractEffect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

Centrally orthocomplete effect algebras

Motivated by the theory of Loomis dimension lattices, we generalize the notion of a hull mapping to an arbitrary effect algebra (EA). Using hull mappings, we identify certain special types of elements in an EA, including generalizations of the invariant elements and of the simple elements in a dimension lattice. We introduce and study a new class of effect algebras, called centrally orthocomplete effect algebras (COEAs), satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. We show that COEAs admit a central cover mapping and we develop the basic theory of direct decomposition of COEAs.

Sharply Orthocomplete Effect Algebras

Special types of effect algebras E called sharply dominating and S-dominating were introduced by S. Gudder in [7, 8]. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of E. Namely we prove that in every sharply orthocomplete S-dominating effect algebra E the set of sharp elements and the center of E are complete lattices bifull in E. If an Archimedean atomic lattice effect algebra E is sharply orthocomplete then it is complete.

Type-Decomposition of an Effect Algebra

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.

Common Generalizations of Orthocomplete and Lattice Effect Algebras

Connections between the weak orthocompleteness and the maximality property in effect algebras are presented. It is proved that an orthomodular poset with the maximality property is disjunctive. A characterization of Archimedean weakly orthocomplete effect algebras is given.

On fuzzy hidden variables

A fuzzy system is introduced as a couple ( E , S ) consisting of a σ -orthocomplete effect algebra E and an order-determining set S of σ -additive states on E, which represents a physical system also taking into account the unsharp quantum measurements. Hidden variables and quasi-hidden variables for fuzzy systems are defined in terms of embedding the system into a fuzzy system based on a tribe of fuzzy sets and a fuzzy system based on a σ -MV algebra, respectively. Relations to the existence of a sufficient supply of σ -additive, resp. finitely additive prime states (which generalize the notion of dispersion-free states) are shown. Some Bell-type inequalities are defined and their relations to hidden variables are shown. Some examples are discussed.

Orthocomplete effect algebras

We prove that for every orthocomplete effect algebra E the center of E forms a complete Boolean algebra. As a consequence, every orthocomplete atomic effect algebra is a direct product of irreducible ones.