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The structure group of a non-degenerate effect algebra

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Abstract

A (non-commutative) structure group G(E) is associated to an arbitrary effect algebra E, and a concept of non-degeneracy is introduced. If E is non-degenerate, G(E) has a right invariant partial order, E embeds as an interval into G(E), and the negative cone of G(E) is a self-similar partial L-algebra. In the degenerate case, the possible anomalies are explained. Lattice effect algebras, 2-divisible effect algebras, and the effect algebra of a Hilbert space, are shown to be non-degenerate. As an application, using a strengthened concept of sharpness, the theory of block decomposition is extended to arbitrary effect algebras, and it is shown that the “very sharp” elements always form a sub-effect algebra and an orthomodular poset.

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Acknowledgements

The authors are grateful to three anonymous referees who pointed out several inaccuracies. Special thanks are owed to the editor and a referee who convinced us to leave the classical terminology of blocks and sharpness untouched. So our new concept of very sharp element, which distils the sharpness paradigm from lattice effect algebras to extend it to general effect algebras, does not interfere with classical sharpness and its vital connection to a lot of important concepts in quantum theory.

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  1. Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70550, Stuttgart, Germany

    Carsten Dietzel & Wolfgang Rump

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  1. Carsten Dietzel
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  2. Wolfgang Rump
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Correspondence to Wolfgang Rump.

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Dedicated to B. V. M.

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Dietzel, C., Rump, W. The structure group of a non-degenerate effect algebra. Algebra Univers. 81, 27 (2020). https://doi.org/10.1007/s00012-020-00657-7

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