Fuzzy complete lattices

Fuzzy Sets and Systems

Volumn 160, Issue 16, 2009, Pages 2275-2291

  Zhang, Qiye ^a,b   Xie, Weixian ^c   Fan, Lei ^d  

^a BEIHANG UNIVERSITY   (China)

^b BEIJING UNIVERSITY OF POSTS AND TELECOMMUNICATIONS   (China)

^c BEIJING INSTITUTE OF FASHION TECHNOLOGY   (China)

^d CAPITAL NORMAL UNIVERSITY   (China)

Author keywords

Fuzzy order; Fuzzy powerset operator; L fuzzy complete lattice; L fuzzy complete lattice homomorphism; Tarski fixed point theorem

Indexed keywords

FUZZY ORDER; FUZZY POWERSET OPERATOR; L-FUZZY COMPLETE LATTICE; L-FUZZY COMPLETE-LATTICE-HOMOMORPHISM; TARSKI FIXED-POINT THEOREM;

THEOREM PROVING;

EID: 67349218115     PISSN: 01650114     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.fss.2008.12.001     Document Type: Article

References (48)

1. Abramsky S., and Jung A. Domain theory. In: Gabbay D.M., and Maibaum T.S.E. (Eds). Handbook of Logic in Computer Science Vol. 3 (1995), Oxford University Press, Oxford 1-168
2. Abramsky S., and Vickers S. Quantales, observational logic and process semantics. Mathematical Structures in Computer Science 3 (1993) 161-227
3. Bělohlávek R. Fuzzy Relational Systems, Foundations and Principles (2002), Kluwer Academic/Plenum Publishers, New York
4. Bělohlávek R. Concept lattices and order in fuzzy logic. Annals of Pure and Applied Logic 128 (2004) 277-298
5. Bělohlávek R. Lattice-type fuzzy order is uniquely given by its 1-Cut: proof and consequences. Fuzzy Sets and Systems 123 (2004) 447-458
6. Bonsangue M.M., van Breugel F., and Rutten J.J.M.M. Generalized ultrametric spaces: completion, topology and powerdomains via the Yoneda embedding. Theoretical Computer Science 193 (1998) 1-51
7. Davey B.A., and Priestley H.A. Introduction to Lattices and Order (1990), Cambridge University Press, Cambridge
8. Eilenberg S., and Kelly G.M. Closed categories. Proc. La Jolla Conf. on Categorical Algebra (1966), Springer, Berlin
9. L. Fan, A new approach to quantitative domain theory, Electronic Notes in Theoretic Computer Science 45 (2001) 〈http://www.elsevier.nl/locate/entcs〉.
10. Flagg R.C. Quantale and continuity spaces. Algebra Universalis 37 (1997) 257-276
11. B. Flagg, P. Sünderhauf, K. Wagner, A logical approach to quantitative domain theory, preprint, 1996, submitted for publication.
12. Flagg B., and Kopperman R. Continuity spaces: reconciling domains and metric spaces. Theoretical Computer Science 177 (1997) 111-138
13. Fuentes-González R. Down and up operators associated to fuzzy relations and t-norms: a definition of fuzzy semi-ideals. Fuzzy Sets and Systems 117 (2001) 377-389
14. Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., and Scott D.S. A Compendium of Continuous Lattices (1980), Springer, Berlin, Heidelberg, New York
15. Goguen J.A. L-fuzzy sets. Journal of Mathematical Analysis and Applications 18 (1967) 145-174
16. Hájek P. Metamathematics of Fuzzy Logic (1998), Kluwer, Dordrecht
17. Höhle U. On the fundamentals of fuzzy set theory. Journal of Mathematical Analysis and Applications 201 (1996) 786-826
18. U. Höhle, M-valued sets and sheaves over integral commutative cl-monoids, in: S.E. Rodabaugh, E.P. Klement, U. Höhle (Eds.), Applications of Category Theory to Fuzzy Subsets, Theory and Decision Library: Series B: Mathematical and Statistical Methods, Vol. 14, Kluwer Academic Publishers, Boston, Dordrecht, London, 1992, pp. 33-72 (Chapter 2).
19. U. Höhle, Categorical axioms of neighborhood systems, in: W. Kotzé (Ed.), Quaestiones Mathematicae 20 (1997) 373-414 (Special Issue).
20. Höhle U., and Blanchard N. Partial ordering in L-underdeterminate sets. Information Sciences 35 (1985) 133-144
21. U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999.
22. W. Kotzé (Ed.), Quaestiones Mathematicae 20 (1997) (Special Issue).
23. Lai H.L., and Zhang D.X. Fuzzy preorder and fuzzy topology. Fuzzy Sets and Systems 157 (2006) 1865-1885
24. H.L. Lai, D.X. Zhang, Many-valued complete distributivity, arXiv:math.CT/0603590, 2006.
25. Johnstone P.T. Stone Spaces (1982), Cambridge University Press, Cambridge
26. Kelly G.M. Basic Concepts of Enriched Category Theory, London Mathematical Soceity Lecture Notes Series Vol. 64 (1982), Cambridge University Press, Cambridge (Also: Reprints in Theory and Applications of Categories (10) (2005).)
27. Lawvere F.W. Metric spaces generalized logic, and closed categories. Rendiconti del Seminario Matematico e Fisicodi Milano 43 (1973) 135-166 (Also: Reprints in Theory and Applications of Categories (1) (2002).)
28. Mac Lane S. Categories for the Working Mathematician. 2nd ed. (2003), Springer, Berlin, Heidelberg, New York
29. Manes E.G. Algebraic Theories (1976), Springer, Berlin, Heidelberg, New York
30. Rodabaugh S.E. Point-set lattice-theoretic topology. Fuzzy Sets and Systems 40 (1991) 297-345
31. S.E. Rodabaugh, Powerset operator based foundation for point-set lattice-theoretic (poslat) fuzzy set theories and topologies, in: W. Kotzé (Ed.), Quaestiones Mathematicae 20(Suppl.), (1997) 463-530.
32. S.E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Boston, Dordrecht, London, 1999, pp. 91-116 (Chapter 2).
33. S.E. Rodabaugh, E.P. Klement, U. Höhle (Eds.), Applications of Category Theory to Fuzzy Subsets, Theory and Decision Library: Series B: Mathematical and Statistical Methods, Vol. 14, 1992, Kluwer Academic Publishers, Boston, Dordrecht, London.
34. S.E. Rodabaugh, E.P. Klement (Eds.), Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic, Vol. 20, 2003, Kluwer Academic Publishers, Boston, Dordrecht, London.
35. Rutten J.J.M.M. Elements of generalized ultrametric domain theory. Theoretical Computer Science 170 (1996) 349-381
36. J.J.M.M. Rutten, Weighted colimits and formal balls in generalized metric spaces, CWI, SEN-R9711, 1997.
37. Šešelja B., and Tepavčević A. Fuzzifying closure systems and fuzzy lattices. In: Aijun A. (Ed). Proc. 11th Internat. Conf. on Rough Sets, Fuzzy Sets, Data Mining and Granular Computing (RSFDGrC 2007), Lecture Notes in Computer Science/Lecture Notes in Artificial Intelligence Vol. 4482 (2007), Springer, Berlin 111-118
38. Šešelja B., and Tepavčević A. Fuzzy ordering relation and fuzzy poset. In: Ashish G., Rajat K.D., and Sankar K.P. (Eds). Proc. Second Internat. Conf. on Pattern Recognition and Machine Intelligence (PReMI 2007), Lecture Notes in Computer Science Vol. 4815 (2007), Springer, Berlin 209-216
39. L.N. Stout, Fully fuzzy topology, in: S.E. Rodabaugh, E.P. Klement (Eds.), Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets, Trends in Logic, Vol. 20, 2003, Kluwer Academic Publishers, Boston, Dordrecht, London, pp. 235-253 (Chapter 8).
40. Tepavčević A., and Trajkovski G. L-fuzzy lattices: an introduction. Fuzzy Sets and Systems 123 (2001) 209-216
41. K.R. Wagner, Solving recursive domain equations with enriched categories, Ph.D. Thesis, Carnegie Mellon University, Technical Report CMU-CS-94-159, July 1994.
42. Wagner K.R. Liminf convergence in Ω-categories. Theoretical Computer Science 184 (1997) 61-104
43. Zadeh L.A. Fuzzy sets. Information and Control 8 (1965) 338-353
44. Zadeh L.A. Similarity relations and fuzzy orderings. Information Sciences 3 (1971) 159-176
45. Zhang Q.Y., and Fan L. Continuity in quantitative domains. Fuzzy Sets and Systems 154 (2005) 118-131
46. Q.Y. Zhang, L. Fan, A kind of L-fuzzy complete lattices and adjoint functor theorem for LF-Posets, Report on the Fourth International Symposium on Domain Theory, Hunan University, Changsha, China, June 2006.
47. W.X. Xie, Q.Y. Zhang, L. Fan, The Dedekind-MacNeille completions for fuzzy posets, Fuzzy Sets and Systems, in press, doi:10.1016/j.fss.2008.12.002.
48. Q.Y. Zhang, L. Fan, W.X. Xie, Adjoint functor theorem for L-fuzzy posets, Bulletin of the Allahabad Mathematical Society, submitted for publication.