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Zhang, Qiye; Xie, Weixian; Fan, Lei

Fuzzy complete lattices. (English) Zbl 1183.06004

Fuzzy Sets Syst. 160, No. 16, 2275-2291 (2009).
The authors present an approach to fuzzification of complete lattices, which is a special kind of complete \(\Omega\)-lattices defined by Lai and Zhang. Tarski fixed-point theorem for the \(L\)-fuzzy complete lattices was proved in a different way. Some fuzzy powerset operators are suggested.
Reviewer: Fu-Gui Shi (Beijing)

Cited in 33 Documents

MSC:

06B23 Complete lattices, completions
06D72 Fuzzy lattices (soft algebras) and related topics
18B35 Preorders, orders, domains and lattices (viewed as categories)

Keywords:

\(L\)-fuzzy complete lattice; Tarski fixed-point theorem; fuzzy powerset operator; \(L\)-fuzzy complete-lattice-homomorphism
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\textit{Q. Zhang} et al., Fuzzy Sets Syst. 160, No. 16, 2275--2291 (2009; Zbl 1183.06004)
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References:

[1] Abramsky, S.; Jung, A., Domain theory, (), 1-168
[2] Abramsky, S.; Vickers, S., Quantales, observational logic and process semantics, Mathematical structures in computer science, 3, 161-227, (1993) · Zbl 0823.06011
[3] Bělohlávek, R., Fuzzy relational systems, foundations and principles, (2002), Kluwer Academic/Plenum Publishers New York · Zbl 1067.03059
[4] Bělohlávek, R., Concept lattices and order in fuzzy logic, Annals of pure and applied logic, 128, 277-298, (2004) · Zbl 1060.03040
[5] Bělohlávek, R., Lattice-type fuzzy order is uniquely given by its 1-cut: proof and consequences, Fuzzy sets and systems, 123, 447-458, (2004) · Zbl 1044.06002
[6] Bonsangue, M.M.; van Breugel, F.; Rutten, J.J.M.M., Generalized ultrametric spaces: completion, topology and powerdomains via the yoneda embedding, Theoretical computer science, 193, 1-51, (1998) · Zbl 0997.54042
[7] Davey, B.A.; Priestley, H.A., Introduction to lattices and order, (1990), Cambridge University Press Cambridge · Zbl 0701.06001
[8] Eilenberg, S.; Kelly, G.M., Closed categories, () · Zbl 0192.10604
[9] L. Fan, A new approach to quantitative domain theory, Electronic Notes in Theoretic Computer Science 45 (2001) \(\langle\)http://www.elsevier.nl/locate/entcs⟩. · Zbl 1260.68217
[10] Flagg, R.C., Quantale and continuity spaces, Algebra universalis, 37, 257-276, (1997) · Zbl 0906.54024
[11] B. Flagg, P. Sünderhauf, K. Wagner, A logical approach to quantitative domain theory, preprint, 1996, submitted for publication.
[12] Flagg, B.; Kopperman, R., Continuity spaces: reconciling domains and metric spaces, Theoretical computer science, 177, 111-138, (1997) · Zbl 0901.68109
[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, 377-389, (2001) · Zbl 0965.03065
[14] Gierz, G.; Hofmann, K.H.; Keimel, K.; Lawson, J.D.; Mislove, M.; Scott, D.S., A compendium of continuous lattices, (1980), Springer Berlin, Heidelberg, New York · Zbl 0452.06001
[15] Goguen, J.A., L-fuzzy sets, Journal of mathematical analysis and applications, 18, 145-174, (1967) · Zbl 0145.24404
[16] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030
[17] Höhle, U., On the fundamentals of fuzzy set theory, Journal of mathematical analysis and applications, 201, 786-826, (1996) · Zbl 0860.03038
[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). · Zbl 0891.54002
[20] Höhle, U.; Blanchard, N., Partial ordering in L-underdeterminate sets, Information sciences, 35, 133-144, (1985) · Zbl 0576.06004
[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.; Zhang, D.X., Fuzzy preorder and fuzzy topology, Fuzzy sets and systems, 157, 1865-1885, (2006) · Zbl 1118.54008
[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 · Zbl 0499.54001
[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, 135-166, (1973), (Also: Reprints in Theory and Applications of Categories (1) (2002).) · Zbl 0335.18006
[28] Mac Lane, S., Categories for the working Mathematician, (2003), Springer Berlin, Heidelberg, New York · Zbl 0906.18001
[29] Manes, E.G., Algebraic theories, (1976), Springer Berlin, Heidelberg, New York · Zbl 0489.18003
[30] Rodabaugh, S.E., Point-set lattice-theoretic topology, Fuzzy sets and systems, 40, 297-345, (1991) · Zbl 0733.54003
[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. · Zbl 0911.04003
[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). · Zbl 0974.03047
[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. · Zbl 0741.00078
[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. · Zbl 1020.00006
[35] Rutten, J.J.M.M., Elements of generalized ultrametric domain theory, Theoretical computer science, 170, 349-381, (1996) · Zbl 0874.68189
[36] J.J.M.M. Rutten, Weighted colimits and formal balls in generalized metric spaces, CWI, SEN-R9711, 1997. · Zbl 0982.54029
[37] Šešelja, B.; Tepavčević, A., Fuzzifying closure systems and fuzzy lattices, (), 111-118
[38] Šešelja, B.; Tepavčević, A., Fuzzy ordering relation and fuzzy poset, (), 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.; Trajkovski, G., L-fuzzy lattices: an introduction, Fuzzy sets and systems, 123, 209-216, (2001) · Zbl 1009.06007
[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 \(\Omega\)-categories, Theoretical computer science, 184, 61-104, (1997) · Zbl 0935.18008
[43] Zadeh, L.A., Fuzzy sets, Information and control, 8, 338-353, (1965) · Zbl 0139.24606
[44] Zadeh, L.A., Similarity relations and fuzzy orderings, Information sciences, 3, 159-176, (1971) · Zbl 0218.02057
[45] Zhang, Q.Y.; Fan, L., Continuity in quantitative domains, Fuzzy sets and systems, 154, 118-131, (2005) · Zbl 1080.06007
[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. · Zbl 1185.06003
[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. · Zbl 1180.06004
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