Abstract
We show that in any generalized effect algebra (G;⊕, 0) a maximal pairwise summable subset is a sub-generalized effect algebra of (G;⊕, 0), called a summability block. If G is lattice ordered, then every summability block in G is a generalized MV-effect algebra. Moreover, if every element of G has an infinite isotropic index, then G is covered by its summability blocks, which are generalized MV-effect algebras in the case that G is lattice ordered. We also present the relations between summability blocks and compatibility blocks of G. Counterexamples, to obtain the required contradictions in some cases, are given.
Highlights
Introduction and some basic definitionsIn a Hilbert space formalization of quantum mechanics, G
Birkhoff and J. von Neumann proposed the concept of quantum logics
The prototype for the axiomatic system of effect algebras was the set E(H) of all positive linear operators dominated by the identity operator in a Hilbert space
Summary
In a Hilbert space formalization of quantum mechanics, G. Operator representations of abstract effect algebras (i.e. their isomorphism with sub-effect algebras of the standard effect algebra E(H) mentioned above) have been studied It was proved in [14] that the set VD(H) of all positive linear operators in an infinite-dimensional complex Hilbert space H with partially defined sum of operators (which coincides with the usual sum) restricted to the common domains of operators forms a generalized effect algebra. A partial algebra (E; ⊕, 0) is called a generalized effect algebra if 0 ∈ E is a distinguished element and ⊕ is a partially defined binary operation on E which satisfies the following conditions for any x, y, z ∈ E:. In every (generalized) effect algebra E relation ≤ and the partial binary operation can be defined by (PO) x ≤ y iff there exists z ∈ E such that x⊕z = y In that case, such element z is unique and we set z = y x. A lattice effect algebra possessing a unique block is called an MV-effect algebra ( a ↔ b for all a, b ∈ E)
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