Journal of Applied Mathematics

Fuzzy Bases of Fuzzy Domains

Sanping Rao and Qingguo Li

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This paper is an attempt to develop quantitative domain theory over frames. Firstly, we propose the notion of a fuzzy basis, and several equivalent characterizations of fuzzy bases are obtained. Furthermore, the concept of a fuzzy algebraic domain is introduced, and a relationship between fuzzy algebraic domains and fuzzy domains is discussed from the viewpoint of fuzzy basis. We finally give an application of fuzzy bases, where the image of a fuzzy domain can be preserved under some special kinds of fuzzy Galois connections.

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J. Appl. Math., Volume 2013 (2013), Article ID 427250, 10 pages.

First available in Project Euclid: 14 March 2014

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Rao, Sanping; Li, Qingguo. Fuzzy Bases of Fuzzy Domains. J. Appl. Math. 2013 (2013), Article ID 427250, 10 pages. doi:10.1155/2013/427250.

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  • D. S. Scott, “Outline of a mathematical theory of computation,” in Proceedings of the 4th Annual Princeton Conference on Information Sciences and Systems, pp. 169–176, Princeton University Press, Princeton, NJ, USA, 1970.
  • D. Scott, “Continuous lattices,” in Toposes, Algebraic Geometry and Logic, vol. 274 of Lecture Notes in Mathematics, pp. 97–136, Springer, Berlin, Germany, 1972.
  • G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains, vol. 93 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, Mass, USA, 2003.
  • J. J. M. M. Rutten, “Elements of generalized ultrametric domain theory,” Theoretical Computer Science, vol. 170, no. 1-2, pp. 349–381, 1996.
  • B. Flagg, P. Sünderhauf, and K. Wagner, “A Logical Approach to Quantitative Domain Theory, Topology Atlas Preprint no. 23,” 1996,
  • K. R. Wager, Solving recursive domain equations with enriched categories [Ph.D. thesis], School of Computer Science; Carnegie-Mellon University, Pittsburgh, Pa, USA, 1994.
  • Q.-Y. Zhang and L. Fan, “Continuity in quantitative domains,” Fuzzy Sets and Systems, vol. 154, no. 1, pp. 118–131, 2005.
  • W. Yao, “Quantitative domains via fuzzy sets: Part I: continuity of fuzzy directed complete posets,” Fuzzy Sets and Systems, vol. 161, no. 7, pp. 973–987, 2010.
  • W. Yao and F.-G. Shi, “Quantitative domains via fuzzy sets: Part II: fuzzy Scott topology on fuzzy directed-complete posets,” Fuzzy Sets and Systems, vol. 173, pp. 60–80, 2011.
  • D. Hofmann and P. Waszkiewicz, “A duality of quantale-enriched categories,” Journal of Pure and Applied Algebra, vol. 216, no. 8-9, pp. 1866–1878, 2012.
  • D. Hofmann and P. Waszkiewicz, “Approximation in quantale-enriched categories,” Topology and its Applications, vol. 158, no. 8, pp. 963–977, 2011.
  • P. Waszkiewicz, “On domain theory over Girard quantales,” Fundamenta Informaticae, vol. 92, no. 1-2, pp. 169–192, 2009.
  • H. Lai and D. Zhang, “Many-valued completečommentComment on ref. [9?]: Please update the information of this reference, if possible. distributivity,”
  • H. Lai and D. Zhang, “Complete and directed complete $\Omega $-categories,” Theoretical Computer Science, vol. 388, no. 1–3, pp. 1–10, 2007.
  • I. Stubbe, “Categorical structures enriched in a quantaloid: tensored and cotensored categories,” Theory and Applications of Categories, vol. 16, no. 14, pp. 283–306, 2006.
  • I. Stubbe, “Towards dynamic domains: totally continuous cocomplete $\mathcal{Q}$-categories,” Theoretical Computer Science, vol. 373, no. 1-2, pp. 142–160, 2007.
  • G. Birkhoff, Lattice Theory, vol. 25, American Mathematical Society; American Mathematical Society Colloquium Publications, Providence, RI, USA, 3rd edition, 1967.
  • C. Y. Zheng, L. Fan, and H. Cui, Frame and Continuous Lattices, Capital Normal University Press, Beijing, China, 2nd edition, 2000 (Chinese).
  • R. Bělohlávek, “Fuzzy Galois connections,” Mathematical Logic Quarterly, vol. 45, no. 4, pp. 497–504, 1999.
  • R. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic Publishers; Plenum Publishers, New York, NY, USA, 2002.
  • L. Fan, “A new approach to quantitative domain theory,” Electronic Notes in Theoretical Computer Science, vol. 45, pp. 77–87, 2001.
  • X. Ma, J. Zhan, and W. A. Dudek, “Some kinds of (e, e or q)-fuzzy filters of BL-algebras,” Computers & Mathematics with Applications, vol. 58, no. 2, pp. 248–256, 2009.
  • J. Zhan and Y. B. Jun, “Soft BL-algebras based on fuzzy sets,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2037–2046, 2010.
  • Q. Zhang, W. Xie, and L. Fan, “Fuzzy complete lattices,” Fuzzy Sets and Systems, vol. 160, no. 16, pp. 2275–2291, 2009.
  • W. Yao and L.-X. Lu, “Fuzzy Galois connections on fuzzy posets,” Mathematical Logic Quarterly, vol. 55, no. 1, pp. 105–112, 2009.