Completely lattice \(L\)-ordered sets with and without \(L\)-equality.

*(English)*Zbl 1218.06003Summary: 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.

##### Keywords:

fuzzy equality; fuzzy order; \(L\)-ordered set; fuzzy concept lattice; Dedekind-MacNeille completion
PDF
BibTeX
XML
Cite

\textit{P. Martinek}, Fuzzy Sets Syst. 166, No. 1, 44--55 (2011; Zbl 1218.06003)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Běhounek, L.; Bodenhofer, U.; Cintula, P., Relations in fuzzy class theory: initial steps, Fuzzy sets and systems, 159, 1729-1772, (2008) · Zbl 1185.03078 |

[2] | Bělohlávek, R., Fuzzy Galois connections, Math. logic Q., 45, 4, 497-504, (1999) · Zbl 0938.03079 |

[3] | Bělohlávek, R., Fuzzy relational systems: foundations and principles, (2002), Kluwer New York · Zbl 1067.03059 |

[4] | Bělohlávek, R., Concept lattices and order in fuzzy logic, Ann. pure appl. logic, 128, 1-3, 277-298, (2004) · Zbl 1060.03040 |

[5] | 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 |

[6] | Bodenhofer, U., Representations and constructions of similarity-based fuzzy orderings, Fuzzy sets and systems, 137, 113-136, (2003) · Zbl 1052.91032 |

[7] | Coppola, C.; Gerla, G.; Pacelli, T., Convergence and fixed points by fuzzy orders, Fuzzy sets and systems, 159, 1178-1190, (2008) · Zbl 1176.06002 |

[8] | Davey, B.A.; Priestley, H.A., Introduction to lattices and order, (2002), Cambridge University Press Cambridge · Zbl 1002.06001 |

[9] | 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 |

[10] | 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 |

[11] | 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 |

[12] | 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 |

[13] | Goguen, J.A., L-fuzzy sets, J. math. anal. appl., 18, 145-174, (1967) · Zbl 0145.24404 |

[14] | Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030 |

[15] | Höhle, U.; Blanchard, N., Partial ordering in L-underdeterminate sets, Inf. sci., 35, 133-144, (1985) · Zbl 0576.06004 |

[16] | Höhle, U., On the fundamentals of fuzzy set theory, J. math. anal. appl., 201, 786-826, (1996) · Zbl 0860.03038 |

[17] | Höhle, U., Many-valued equalities, singletons and fuzzy partitions, Soft comput., 2, 134-140, (1998) |

[18] | Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall Upper Saddle River · Zbl 0915.03001 |

[19] | H.L. Lai, D.X. Zhang, Many-valued complete distributivity, arXiv:math.CT/0603590, 2006. |

[20] | 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. |

[21] | Ovchinnikov, S.V., Structure of fuzzy binary relations, Fuzzy sets and systems, 6, 169-195, (1981) · Zbl 0464.04004 |

[22] | Venugopalan, P., Fuzzy ordered sets, Fuzzy sets and systems, 46, 221-226, (1992) · Zbl 0765.06008 |

[23] | Xie, W.; Zhang, Q.; Fan, L., The dedekind – macneille completions for fuzzy posets, Fuzzy sets and systems, 160, 2292-2316, (2009) · Zbl 1185.06003 |

[24] | Yao, W.; Lu, L.-X., Fuzzy Galois connections on fuzzy posets, Math. logic Q., 55, 1, 105-112, (2009) · Zbl 1172.06001 |

[25] | 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 |

[26] | Zadeh, L.A., Similarity relations and fuzzy orderings, Inf. sci., 3, 177-200, (1971) · Zbl 0218.02058 |

[27] | Zhang, Q.Y.; Fan, L., Continuity in quantitative domains, Fuzzy sets and systems, 154, 118-131, (2005) · Zbl 1080.06007 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.