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Girard Couples of Quantales

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Abstract

We introduce the concept of a Girard couple, which consists of two (not necessarily unital) quantales linked by a strong form of duality. The two basic examples of Girard couples arise in the study of endomorphism quantales and of the spectra of operator algebras. We construct, for an arbitrary sup-lattice S, a Girard quantale whose right-sided part is isomorphic to S.

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References

  1. Barr, M.: ∗-autonomous categories. Lecture Notes in Mathematics, vol. 752. Springer, Berlin (1979). With an appendix by Po Hsiang Chu

    Google Scholar 

  2. Chang, C.C.: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88, 467–490 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cockett, J.R.B., Seely, R.A.G.: Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories. Theory Appl. Categ. 3(5), 85–131 (1997) (electronic)

    MATH  MathSciNet  Google Scholar 

  4. Egger, J.M.: *-autonomous functor categories. Theory Appl. Categ. 20 (2008, in press)

  5. Girard, J.-Y.: Linear logic. Theoret. Comput. Sci. 50(1), 101 (1987)

    Article  MathSciNet  Google Scholar 

  6. Joyal, A., Tierney, M.: An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society, vol. 309 (1984)

  7. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras I, II. Graduate Studies in Mathematics, vol. 15, 16. American Mathematical Society (1997)

  8. Kruml, D., Paseka, J.: Algebraic and categorical aspects of quantales. In: Handbook of Algebra, vol. 3, pp. 1–39. Elsevier (2008)

  9. Mulvey, C.J., Pelletier, J.W.: A quantisation of the calculus of relations. In: Canadian Mathematical Society Conference Proceedings, vol. 13, pp. 345–360. American Mathematical Society, Providence (1992)

    Google Scholar 

  10. Mulvey, C.J., Pelletier, J.W.: On the quantisation of points. J. Pure Appl. Algebra 159, 231–295 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Paoli, F.: A common abstraction of MV-algebras and abelian l-groups. Stud. Log. 65(3), 355–366 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Paseka, J.: Simple quantales. In: Proceedings of the 8th Prague Topological Symposium, pp. 314–428. Topology Atlas (1997)

  13. Pelletier, J.W.: Von Neumann algebras and Hilbert quantales. Appl. Categ. Struct. 5, 249–264 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Pelletier, J.W., Rosický, J.: Simple involutive quantales. J. Algebra 195, 367–386 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Raney, G.N.: Tight Galois connections and complete distributivity. Trans. Amer. Math. Soc. 97, 418–426 (1960)

    Article  MathSciNet  Google Scholar 

  16. Rosenthal, K.I.: Quantales and their applications. Pitman Research Notes in Mathematics Series, vol. 234. Longman, Essex (1990)

    MATH  Google Scholar 

  17. Yetter, D.N.: Quantales and (noncommutative) linear logic. J. Symb. Log. 55(1), 41–64 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Kotlářská 2, Brno, 60200, Czech Republic

    J. M. Egger & David Kruml

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  1. J. M. Egger
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  2. David Kruml
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Correspondence to David Kruml.

Additional information

The second author is supported by the Ministry of Education of the Czech Republic under the project MSM143100009.

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Egger, J.M., Kruml, D. Girard Couples of Quantales. Appl Categor Struct 18, 123–133 (2010). https://doi.org/10.1007/s10485-008-9138-3

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  • DOI: https://doi.org/10.1007/s10485-008-9138-3

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