Representations of orthomodular structures
by
Josef Tkadlec
Czech Technical University

There are various possibilities how to represent orthomodular structures. One attempt (topological) follows the famous Stone construction for representation of Boolean algebras and leads to a representation by means of clopen subsets of a closure space . This enables (as we will show) to visualize some algebraic constructions and proofs.

Another attempt (graphical) pays attention to orthogonality relations and leads to orthogonality diagrams , more compact Greechie diagrams  and their generalizations  and to dual diagrams. These representations are useful especially if we study probability measures (states). We will give several examples for so-called Kochen-Specker type constructions , i.e., orthomodular lattices realizable in a Hilbert space with a small (empty be a special case of smallness) set of two-valued states.

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Greechie, R. J.: Orthomodular lattices admitting no states. J. Combin. Theory Ser. A 10 (1971), 119-132.

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Kochen, S., Specker, E. P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17 (1967), 59-87.

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Navara, M., Rogalewicz, V.: The pasting constructions for orthomodular posets. Math. Nachrichten 154 (1991), 157-168.

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Tkadlec, J.: Partially additive measures and set representations of orthoposets. J. Pure Appl. Algebra 86 (1993), 79-94.

Date received: April 30, 1998

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Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caba-10.