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Completely lattice $$L$$-ordered sets with and without $$L$$-equality. (English) Zbl 1218.06003
Summary: A relationship between $$L$$-order based on an $$L$$-equality and $$L$$-order based on crisp equality is explored in detail. This enables us to clarify some properties of completely lattice $$L$$-ordered sets and generalize some related assertions. Namely, Bělohlávek’s main theorem of fuzzy concept lattices is generalized as well as his theorem dealing with Dedekind-MacNeille completion. Analogously, completion of an $$L$$-ordered set via the completely lattice $$L$$-ordered set of all down-$$L$$-sets is described.

##### MSC:
 06A75 Generalizations of ordered sets 06B75 Generalizations of lattices
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##### References:
  Běhounek, L.; Bodenhofer, U.; Cintula, P., Relations in fuzzy class theory: initial steps, Fuzzy sets and systems, 159, 1729-1772, (2008) · Zbl 1185.03078  Bělohlávek, R., Fuzzy Galois connections, Math. logic Q., 45, 4, 497-504, (1999) · Zbl 0938.03079  Bělohlávek, R., Fuzzy relational systems: foundations and principles, (2002), Kluwer New York · Zbl 1067.03059  Bělohlávek, R., Concept lattices and order in fuzzy logic, Ann. pure appl. logic, 128, 1-3, 277-298, (2004) · Zbl 1060.03040  Bodenhofer, U., A similarity-based generalization of fuzzy orderings preserving the classical axioms, Int. J. uncertain. fuzziness knowl. based syst., 8, 5, 593-610, (2000) · Zbl 1113.03333  Bodenhofer, U., Representations and constructions of similarity-based fuzzy orderings, Fuzzy sets and systems, 137, 113-136, (2003) · Zbl 1052.91032  Coppola, C.; Gerla, G.; Pacelli, T., Convergence and fixed points by fuzzy orders, Fuzzy sets and systems, 159, 1178-1190, (2008) · Zbl 1176.06002  Davey, B.A.; Priestley, H.A., Introduction to lattices and order, (2002), Cambridge University Press Cambridge · Zbl 1002.06001  Demirci, M., A theory of vague lattices based on many-valued equivalence relations—I: general representation results, Fuzzy sets and systems, 151, 437-472, (2005) · Zbl 1067.06006  Demirci, M., A theory of vague lattices based on many-valued equivalence relations—II: complete lattices, Fuzzy sets and systems, 151, 473-489, (2005) · Zbl 1067.06007  L. Fan, A new approach to quantitative domain theory, in: Electronic Notes in Theoretical Computer Science, vol. 45, 2001, pp. 77-87. · Zbl 1260.68217  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  Goguen, J.A., L-fuzzy sets, J. math. anal. appl., 18, 145-174, (1967) · Zbl 0145.24404  Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030  Höhle, U.; Blanchard, N., Partial ordering in L-underdeterminate sets, Inf. sci., 35, 133-144, (1985) · Zbl 0576.06004  Höhle, U., On the fundamentals of fuzzy set theory, J. math. anal. appl., 201, 786-826, (1996) · Zbl 0860.03038  Höhle, U., Many-valued equalities, singletons and fuzzy partitions, Soft comput., 2, 134-140, (1998)  Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall Upper Saddle River · Zbl 0915.03001  H.L. Lai, D.X. Zhang, Many-valued complete distributivity, arXiv:math.CT/0603590, 2006.  P. Martinek, On generalization of fuzzy concept lattices based on change of underlying fuzzy order, in: R. Belohlavek, S.O. Kuznetsov (Eds.), CLA 2008, Proceedings of the Sixth International Conference on Concept Lattices and their Applications, Palacky University, Olomouc, 2008, pp. 207-215.  Ovchinnikov, S.V., Structure of fuzzy binary relations, Fuzzy sets and systems, 6, 169-195, (1981) · Zbl 0464.04004  Venugopalan, P., Fuzzy ordered sets, Fuzzy sets and systems, 46, 221-226, (1992) · Zbl 0765.06008  Xie, W.; Zhang, Q.; Fan, L., The dedekind – macneille completions for fuzzy posets, Fuzzy sets and systems, 160, 2292-2316, (2009) · Zbl 1185.06003  Yao, W.; Lu, L.-X., Fuzzy Galois connections on fuzzy posets, Math. logic Q., 55, 1, 105-112, (2009) · Zbl 1172.06001  Yao, W., Quantitative domains via fuzzy sets: part I: continuity of fuzzy directed complete posets, Fuzzy sets and systems, 161, 973-987, (2010) · Zbl 1193.06007  Zadeh, L.A., Similarity relations and fuzzy orderings, Inf. sci., 3, 177-200, (1971) · Zbl 0218.02058  Zhang, Q.Y.; Fan, L., Continuity in quantitative domains, Fuzzy sets and systems, 154, 118-131, (2005) · Zbl 1080.06007
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