# Abstract:

The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e. g., \cite{pp:book,wata}). Recently an effort has been exercised to advance with logics that possess a symmetric difference (\cite{matODL,MP1}) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In \cite{matODL} the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result of this paper somewhat economizes on this construction: There is an ODL with 3 generators that is not set-representable (and so the free ODL with 3 generators cannot be set-representable). The result is based on a specific technique of embedding orthomodular lattices into ODLs. The ODLs with 2 generators are always set-representable as we show by characterizing the free ODL with 2 generators - this ODL is ${\rm MO}_3 \times 2^4$.

# Keywords:

orthomodular lattice, quantum logic, symmetric difference, Gödel's coding, Boolean algebra, free algebra

# Classification:

06C15, 03G12, 81B10

# References:

1. L.~Beran: Orthomodular Lattices, Algebraic Approach. D. Reidel, Dordrecht, 1985.   CrossRef
2. G.~Bruns and J.~Harding: Algebraic aspects of orthomodular lattices. In: Current Research in Operational Quantum Logic (B.~Coecke, D.~Moore and A.~Wilce, eds.), Kluwer Academic Publishers 2000, pp. 37-65.   CrossRef
3. S.~Burris and H. P.~Sankappanavar: A Course in Universal Algebra Springer-Verlag, New York 1981.   CrossRef
4. M. L.~Dalla Chiara, R.~Giuntini and R.~Greechie: Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics. Kluwer Academic Publishers, Dordrecht, Boston, London 2004.   CrossRef
5. G.~Dorfer, A.~Dvure\v{c}enskij and H. M.~L\"{a}nger: Symmetric difference in orthomodular lattices. Math. Slovaca 46 (1996), 435-444.   CrossRef
6. G.~Dorfer: Non-commutative symmetric differences in orthomodular lattices. Internat. J. Theoret. Phys. 44 (2005), 885-896.   CrossRef
7. A.~Dvure\v censkij and S.~Pulmannov\' a: New Trends in Quantum Structures. Kluwer Acad. Publ., Dordrecht, and Ister Science, Bratislava 2000.   CrossRef
8. R.~Godowski and R. J.~Greechie: Some equations related to states on orthomodular lattices. Demonstratio Math. XVII (1984), 1, 241-250.   CrossRef
9. R. J.~Greechie: Orthomodular lattices admitting no states. J. Combinat. Theory 10 (1971), 119-132.   CrossRef
10. K.~Engesser, D.M.~Gabbay and D.~Lehmann ed.: Handbook of Quantum Logic and Quantum Structures. Elsevier 2007.   CrossRef
11. G.~Kalmbach: Orthomodular Lattices. Academic Press, London 1983.   CrossRef
12. M.~Matou\v{s}ek: Orthocomplemented lattices with a symmetric difference. Algebra Universalis 60 (2009), 185-215.   CrossRef
13. M.~Matou\v{s}ek and P.~Pt\'{a}k: Orthocomplemented posets with a symmetric difference. Order 26 (2009), 1-21.   CrossRef
14. M.~Matou\v{s}ek and P.~Pt\'{a}k: On identities in orthocomplemented difference lattices. Math. Slovaca 60 (2010), 5, 583-590.   CrossRef
15. M.~Matou\v{s}ek and P.~Pt\'{a}k: Symmetric difference on orthomodular lattices and $Z_2$-valued states. Comment. Math. Univ. Carolin. 50 (2009), 4, 535-547.   CrossRef
16. M.~Navara and P.~Pt\'{a}k: Almost Boolean orthomodular posets. J. Pure Appl. Algebra 60 (1989), 105-111.   CrossRef
17. E.~Park, M. M.~Kim and J. Y.~Chung: A note on symmetric differences of orthomodular lattices. Commun. Korean Math. Soc. 18 (2003), 2, 207-214.   CrossRef
18. P.~Pt\'{a}k and S.~Pulmannov\'{a}: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht, Boston, London 1991.   CrossRef
19. S.~Watanabe: Modified concepts of logic, probability and integration based on generalized continuous characteristic function. Inform. and Control 2 (1969), 1-21.   CrossRef