Previous |  Up |  Next


affine homeomorphism; compact convex set; hypergraph; unital orthomodular lattices; state space representation; orthomodular lattice; state space
Using the general hypergraph technique developed in [7], we first give a much simpler proof of Shultz's theorem [10]: Each compact convex set is affinely homeomorphic to the state space of an orthomodular lattice. We also present partial solutions to open questions formulated in [10] - we show that not every compact convex set has to be a state space of a unital orthomodular lattice and that for unital orthomodular lattices the state space characterization can be obtained in the context of unital hypergraphs.
[1] Greechie R.J.: Orthomodular lattices admitting no states. J. Comb. Theory 10(1971), 119-132. DOI 10.1016/0097-3165(71)90015-X | MR 0274355 | Zbl 0219.06007
[2] Gudder S.P.: Stochastic Methods in Quantum Mechanics. North Holland, New York, 1979. MR 0543489 | Zbl 0439.46047
[3] Gudder S., Kläy M.P., Rüttimann G.T.: States on hypergraphs. Demonstratio Math. 19 (1986), 503-526. MR 0895021
[4] Kalmbach G.: Orthomodular Lattices. Academic Press, London, 1983. MR 0716496 | Zbl 0528.06012
[5] Navara M.: State space properties of finite logics. Czechoslovak Math. J. 37 (112) (1987), 188-196. MR 0882593 | Zbl 0647.03057
[6] Navara M.: State space of quantum logics. Thesis, Technical University of Prague, 1987. (In Czech.)
[7] Navara M., Rogatewicz V.: Construction of orthomodular lattices with given state spaces. Demonstratio Math. 21 (1988), 481-493. MR 0981700
[8] Pták P.: Exotic logics. Coll. Math. 54 (1987), 1-7. MR 0928651
[9] Pták P., Pulmannová S.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. MR 1176314 | Zbl 0743.03039
[10] Shultz F. W.: A characterization of state spaces of orthomodular lattices. J. Comb. Theory (A) 17 (1974), 317-328. DOI 10.1016/0097-3165(74)90096-X | MR 0364042
Partner of
EuDML logo