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Logics of √’qMV algebras

Logics of √’qMV algebras. Antonio Ledda Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks Università di Cagliari. Siena, September 8 th 2008. Some motivation.

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Logics of √’qMV algebras

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  1. Logics of √’qMV algebras Antonio Ledda Joint work with F. Bou, R. Giuntini, F. Paoli and M. Spinks Università di Cagliari Siena, September 8th 2008

  2. Some motivation qMV algebras were introduced in an attempt to provide a convenient abstraction of the algebra over the set of all density operators of the two-dimensional complex Hilbert space, endowed with a suitable stock of quantum gates.

  3. The definition of qMV-algebra Definition Smoothness axioms Łukasiewicz’s axiom

  4. qMV-algebras

  5. Adding the square root of the negation √’qMV algebras were introduced as term expansions of quasi-MV algebras by an operation of square root of the negation.

  6. Adding the square root of the negation

  7. quasi-Wajsberg algebras

  8. Term equivalence Theorem

  9. The standard Wajsberg algebra St

  10. The algebra F[0,1]

  11. The standard qW algebras S and D

  12. Equationally defined preorder

  13. An example of equationally defined preorder

  14. Logics from equationally preordered classes

  15. Remark

  16. A logic from an equationally preordered variety

  17. The quasi-Łukasiewicz logic qŁ

  18. A remark

  19. Summary of the logic results

  20. A “logical” version ofqMV

  21. Term equivalences

  22. Logics of qMV algebras (1)

  23. Logics of qMV algebras (2)

  24. Logics of qMV algebras (3) Most logics in the previous schema look noteworthy under some respect:

  25. 1-cartesian algebras

  26. Examples

  27. Inclusion relationships

  28. Placing our logics in the Leibniz hierarchy (1) Well-behaved logics • is regularly algebraisable and is its equivalent quasivariety semantics; • is regularly algebraisable and is its equivalent quasivariety semantics; (they are the 1-assertional logics of relatively 1-regular quasivarieties)

  29. Placing our logics in the Leibniz hierarchy (2) Ill-behaved logics None of the other logics is protoalgebraic: • : the Leibniz operator is not monotone on the deductive filters of F120; • : it is a sublogic of such; • : the Leibniz operator is not monotone on the deductive filters of ;

  30. Placing our logics in the Leibniz hierarchy (2)

  31. Placing our logics in the Frege hierarchy

  32. Some notations We use the following abbreviations:

  33. The logics C and C1

  34. The logics C and C1

  35. A completeness result

  36. The notion of (strong) implicative filter

  37. Remark In the definition of strong implicative filter conditions F2, F3, F4, F5 are redundant

  38. A characterization of the deductive filters

  39. Thank you for your attention!!

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