3- and 4-dim Greechie diagrams and their corresponding Hasse diagrams shown above them (Ref. 56) (Fig. 18, p. 84).
(a) OML L42 (Ref. 17) in which Eq. (16) fails; (b) (Ref. 19) in which Eq. (16) fails; (c) (Ref. 19) in which Eq. (17) fails.
KS original setup (Ref. 23) How to turn their figure into a MMP is in Ref. 58.
Peres’ KS MMP hypergraph.
Bub’s MMP with 49 atoms and 36 blocks. Notice that 12 bigger dots with a pattern (red online) represent just four atoms: 4, 5, 6, and .
Conway–Kochen’s MMP. Notice that we cannot drop blocks containing , , 7, and G because atoms 4, 5, 6, and , from them also share two other blocks each. Why we cannot drop atoms 7, G, H, J, etc., is explained in the text.
(a) Greechie diagrams that correspond to modular lattices with 1one to four blocks (over each other); (b) their vectors triples; (c) the smallest nonmodular lattice; (d) its vector triples.
A new kind of lattice (MMPL) in which Bub’s setup passes 3OA. The inequality relation in Eq. (36) is represented by the thick red line.
(a) A 13–7 OML that admits a strong set of states but violates Hilbert-space orthoarguesian equations; (b) a 16–9 OML that does not admit a strong set of states but satisfies orthoarguesian equations; (c) a 16–10 OML that admits neither a strong nor a full set of states and violates all orthoarguesian equations.
Starting configuration for generation of 41–41 MMPs.
36–36 OML that admits exactly one state and is dual to itself. It is given in the standard compact representation in the first figure and in our separate cycle representation in the other three figures. The figures are explained in detail in Sec. VII.
First figure shows a 35–35 lattice presented by means of its biggest loop, hexadecagon; it exhibits a left-right symmetry with respect to an axis through vertices V and Y. Three other figures show the same OML in the separate cycle representation. They are explained in detail in Sec. VII.
39–39–06 OML dual to itself.
40–40–34 OML dual to itself.
40–40–38 OML dual to itself.
Summary of known equations holding in (quantum) Hilbert lattices
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