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L-quantum spaces

L-量子空间

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Abstract

In this paper, based on a complete residuated lattice L, we introduce the definitions of L-quantum spaces and continuous mappings. Then we establish an adjunction between the category of stratified L-quantum spaces and the opposite category of two-sided L-quantales. We also prove that the category of sober L-quantum spaces is dually equivalent to the category of spatial two-sided L-quantales.

摘要

创新点

  1. (1)

    本文将量化的思想应用于量子空间, 提出了L-量子空间概念, 它可以看作是非交换的L-拓扑空间。

  2. (2)

    给出了L-量子空间上连续映射的概念, 探讨了它的一些性质。

  3. (3)

    研究了L-量子空间的一些范畴性质, 比如 : L-量子空间范畴可以与双侧L-quantale范畴之间建立伴随。

  4. (4)

    得到了Sober L-量子空间范畴的一个等价范畴, 即空间式的双侧L-quantale范畴。

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References

  1. Stone M H. The theory of representations for Boolean algebras. Trans American Math Soc, 1936, 40: 37–111

    MathSciNet  MATH  Google Scholar 

  2. Abramsky S. Domain theory in logical form. Ann Pure Appl Logic, 1991, 51: 1–77

    Article  MathSciNet  MATH  Google Scholar 

  3. Isbell J. Atomless parts of spaces. Math Scand, 1972, 31: 5–32

    MathSciNet  MATH  Google Scholar 

  4. Yao W. An approach to fuzzy frames via fuzzy posets. Fuzzy Sets Syst, 2011, 166: 75–89

    Article  MathSciNet  MATH  Google Scholar 

  5. Birkhoff G, von Neumann J. The logic of quantum mechanics. Ann Math, 1936, 379: 823–843

    Article  MathSciNet  MATH  Google Scholar 

  6. Ying M S. A theory of computation based on quantum logic (I). Theor Comput Sci, 2005, 344: 134–207

    Article  MathSciNet  MATH  Google Scholar 

  7. Ying M S. Quantum logic and automata theory. In: Dov G, Daniel L, Kurt E, eds. Handbook of Quantum Logic and Quantum Structures. North-Holland: Elsevier, 2007. 619–754

    Chapter  Google Scholar 

  8. Shang Y, Lu X, Lu R Q. A theory of computation based on unsharp quantum logic: finite state automata and pushdown automata. Theor Comput Sci, 2012, 434: 53–86

    Article  MathSciNet  MATH  Google Scholar 

  9. Mulvey C J. Suppl Rend Circ Mat Palermo, 1986, 12: 99–104

    MathSciNet  Google Scholar 

  10. Abramsky S, Vickers S. Quantale, observational logic and process semantics. Math Struct Comput Sci, 1993, 3: 161–227

    Article  MathSciNet  MATH  Google Scholar 

  11. Resende P. Quantales, finite observations and strong bisimulation. Theor Comput Sci, 2001, 254: 95–149

    Article  MathSciNet  MATH  Google Scholar 

  12. Li Y M, Li Zh H. Quantales and process semantics of bisimulation. Acta Mathe Sin Chinese Ser, 1999, 42: 313–320

    MathSciNet  MATH  Google Scholar 

  13. Girard J Y. Linear logic. Theor Comput Sci, 1987, 50: 1–102

    Article  MathSciNet  MATH  Google Scholar 

  14. Yetter D. Quantales and noncommutative linear logic. J Symb Logic, 1990, 55: 41–64

    Article  MathSciNet  MATH  Google Scholar 

  15. Parikh R. Some applications of topology to program semantics. Math Syst Theory, 1983, 16: 111–131

    Article  MathSciNet  MATH  Google Scholar 

  16. Borceux F, Bossche G V. An essay on non-commutative topology. Topol Appl, 1989, 31: 203–223

    Article  MATH  Google Scholar 

  17. He W, Luo M K. Quantum spaces. Acta Math Sin English Ser, 2010, 26: 1323–1330

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang K Y. Some researches on fuzzy domains and fuzzy quantales. Dissertation for Ph.D. Degree. Xi’an: Shaanxi Normal University, 2012

    Google Scholar 

  19. Adámek J, Herrlich H, Strecker G E. Abstract and Concrete Categories. New York: Wiley, 1990

    MATH  Google Scholar 

  20. Bělohávek R. Fuzzy Relational Systems: Foundations and Principles. New York: Kluwer Academic Publishers, 2002

    Book  MATH  Google Scholar 

  21. Ward M, Dilworth R P. Residuated lattices. Trans American Math Soc, 1939, 45: 335–353

    Article  MathSciNet  MATH  Google Scholar 

  22. Bělohávek R. Some properties of residuated lattices. Czechoslovak Math J, 2003, 53: 161–171

    Article  MathSciNet  Google Scholar 

  23. Hájek P. Metamathematics of Fuzzy Logic. Dordrecht: Kluwer Academic Publishers, 1998

    Book  MATH  Google Scholar 

  24. Fan L. A new approach to quantitative domain theory. Electron Notes Theor Comput Sci, 2001, 45: 77–87

    Article  MATH  Google Scholar 

  25. Goguen J A. L-fuzzy sets. J Math Anal Appl, 1967, 18: 145–174

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang Q Y, Fan L. Continuity in quantitative domains. Fuzzy Sets Syst, 2005, 154: 118–131

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang Q Y, Xie W X, Fan L. Fuzzy complete lattices. Fuzzy Sets Syst, 2009, 160: 2275–2291

    Article  MathSciNet  MATH  Google Scholar 

  28. Rodabaugh S E. Powerset operator foundations for poslat fuzzy set theories and topologies. In: Höhle U, Rodabaugh S E, eds. Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, the Handbooks of Fuzzy Sets Series, vol.3. Boston, Dordrecht. London: Kluwer Academic Publishers, 1999. 91–116

    Chapter  Google Scholar 

  29. Lai H L, Zhang D X. Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory. Int J Approx Reason, 2009, 50: 695–707

    Article  MathSciNet  MATH  Google Scholar 

  30. Stubbe I. Categorical structures enriched in a quantaloid: tensored and cotensored categories. Theory Appl Categ, 2006, 16: 283–306

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, 710062, Xi’an, China

    Bin Zhao & Nana Ma

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  1. Bin Zhao
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  2. Nana Ma
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Correspondence to Bin Zhao.

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Zhao, B., Ma, N. L-quantum spaces. Sci. China Inf. Sci. 59, 32201 (2016). https://doi.org/10.1007/s11432-015-5416-5

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  • DOI: https://doi.org/10.1007/s11432-015-5416-5

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