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Type-Decomposition of an Effect Algebra

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Abstract

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras.

We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, 1 Sutton Court, Amherst, MA, 01002, USA

    David J. Foulis

  2. Mathematical Institute, Slovak Academy of Sciences, Stefánikova 49, 814 73, Bratislava, Slovakia

    Sylvia Pulmannová

Authors
  1. David J. Foulis
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  2. Sylvia Pulmannová
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Correspondence to David J. Foulis.

Additional information

Dedicated to Professor Dr. Peter Mittelstaedt with admiration for his profound work on quantum logic, quantum measurement theory, and the philosophy of quantum mechanics.

S. Pulmannová was supported by Research and Development Support Agency under the contract No. APVV-0071-06, grant VEGA 2/6088/26 and Center of Excellence SAS, CEPI I/2/2005.

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Foulis, D.J., Pulmannová, S. Type-Decomposition of an Effect Algebra. Found Phys 40, 1543–1565 (2010). https://doi.org/10.1007/s10701-009-9344-3

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