Journal of Applied Mathematics

Fuzzy Bases of Fuzzy Domains

Sanping Rao and Qingguo Li

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807999

Digital Object Identifier
doi:10.1155/2013/427250

Mathematical Reviews number (MathSciNet)
MR3064881

Zentralblatt MATH identifier
1266.06006

Citation

Rao, Sanping; Li, Qingguo. Fuzzy Bases of Fuzzy Domains. J. Appl. Math. 2013 (2013), Article ID 427250, 10 pages. doi:10.1155/2013/427250. https://projecteuclid.org/euclid.jam/1394807999


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References

  • 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, http://at.yorku.ca/e/a/p/p/23.htm.
  • 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,” http://arxiv.org/abs/math/0603590.
  • 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.