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Boolean Subalgebras of Orthoalgebras

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Abstract

We develop a direct method to recover an orthoalgebra from its poset of Boolean subalgebras. For this a new notion of direction is introduced. Directions are also used to characterize in purely order-theoretic terms those posets that are isomorphic to the poset of Boolean subalgebras of an orthoalgebra. These posets are characterized by simple conditions defining orthodomains and the additional requirement of having enough directions. Excepting pathologies involving maximal Boolean subalgebras of four elements, it is shown that there is an equivalence between the category of orthoalgebras and the category of orthodomains with enough directions with morphisms suitably defined. Furthermore, we develop a representation of orthodomains with enough directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph approach extends the technique of Greechie diagrams and resembles projective geometry. Using such hypergraphs, every orthomodular poset can be represented by a set of points and lines where each line contains exactly three points.

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Acknowledgments

C.H. is supported by EPSRC Fellowship EP/L002388/1. B.L. is supported by the AFOSR under the MURI grant number FA9550-16-1-0082 entitled, “Semantics, Formal Reasoning, and Tool Support for Quantum Programming”. M.N. was supported by the Ministry of Education of the Czech Republic under Project RVO13000.

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Authors and Affiliations

  1. New Mexico State University, Las Cruces, NM, USA

    John Harding

  2. University of Edinburgh, Edinburgh, Scotland

    Chris Heunen

  3. Tulane University, New Orleans, LA, 70118, USA

    Bert Lindenhovius

  4. Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague, Czechia

    Mirko Navara

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  1. John Harding
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  2. Chris Heunen
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  3. Bert Lindenhovius
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  4. Mirko Navara
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Correspondence to John Harding.

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Harding, J., Heunen, C., Lindenhovius, B. et al. Boolean Subalgebras of Orthoalgebras. Order 36, 563–609 (2019). https://doi.org/10.1007/s11083-019-09483-6

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