Hull mappings and dimension effect algebras

2012

Algebra Univers.

Self Cite

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 MVeffect algebra.

“…The SK-congruence ∼ on E is called a dimension equivalence relation (DER) [14,Definition 7.14] iff the following additional condition is satisfied:…”

Section: Dimension Effect Algebrasmentioning

confidence: 99%

Section: Finite and Infinite Elementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quotients of dimension effect algebras

2012

Algebra Univers.

Self Cite

“…In this section we give a brief sketch of the decomposition of E into direct sums of various types that are induced by TD subsets of E. Details, proofs, and further related results can be found in [11] and [12].…”

Section: Direct Decomposition Of E Induced By Td Subsetsmentioning

confidence: 99%

Centrally orthocomplete effect algebras

2010

Algebra Univers.

Self Cite

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.

“…The hull mappings featured in the original work of Loomis on dimension lattices [21] were generalized to effect algebras in [11,12]. In the following definition, we further extend the notion of a hull mapping to pseudoeffect algebras.…”

Section: This Proves (Iii)mentioning

confidence: 99%

Type Decomposition of a Pseudoeffect Algebra

Vinceková

2010

J. Aust. Math. Soc.

Self Cite

Effect 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 typedetermining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.2010 Mathematics subject classification: primary 06C15; secondary 17C65, 46L45.