Fuzzy Ordering Relation and Fuzzy Poset
Connections between (weakly) reflexive, antisymmetric and transitive lattice-valued fuzzy relations on a nonempty set X (fuzzy ordering relations on X) and fuzzy subsets of a crisp poset on X (fuzzy posets) are established and various properties of cuts o
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tract. Connections between (weakly) reflexive, antisymmetric and transitive lattice-valued fuzzy relations on a nonempty set X (fuzzy ordering relations on X) and fuzzy subsets of a crisp poset on X (fuzzy posets) are established and various properties of cuts of such structures are proved. A representation of fuzzy sets by cuts corresponding to atoms in atomically generated lattices has also been given. AMS Mathematics Subject Classification (1991): 04A72. Keywords and Phrases: Lattice valued fuzzy ordering relation, fuzzy weak ordering relation, fuzzy poset, cutworthy approach.
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Introduction
The present research belongs to the theory of fuzzy ordering relations. These are investigated as lattice valued structures. Therefore, the framework as well as the subject are within the order theory. Fuzzy structures presented here support cutworthy properties, that is, crisp fuzzified properties are preserved under cut structures. Therefore, the co-domains of all mappings (fuzzy sets) are complete lattices, without additional operations. Namely, such lattices support the transfer of crisp properties to cuts. Due to the extensive research of fuzzy ordering relations, we mention only those authors and papers which are relevant to our approach. From the earlier period, we mention papers by Ovchinnikov ([9,10], and papers cited there); we use his definition of the ordering relation. In the recent period, there are papers by Bˇelohl´avek (most of the relevant results are collected in his book [1]). He investigates also lattice-valued orders, the lattice being residuated. Recent important results concerning fuzzy orders and their representations are obtained by De Baets and Bodenhofer (see a state-of-the-art overview about weak fuzzy orders, [4]).
This research was partially supported by Serbian Ministry of Science and Environment, Grant No. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, grant ”Lattice methods and applications”.
A. Ghosh, R.K. De, and S.K. Pal (Eds.): PReMI 2007, LNCS 4815, pp. 209–216, 2007. c Springer-Verlag Berlin Heidelberg 2007
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ˇ selja and A. Tepavˇcevi´c B. Seˇ
Most of the mentioned investigations of fuzzy orders are situated in the framework of T -norms, or more generally, in the framework of residuated lattices and corresponding operations. As mentioned above, our approach is purely order-theoretic. Motivated by the classical approach to partially ordered sets, we investigate fuzzy orders from two basic aspects. Firstly we consider fuzzy posets, i.e., fuzzy sets on a crisp ordered set. The other aspect is a fuzzification of an ordering relation. As the main result we prove that there is a kind of natural equivalence among these two structures. Namely, starting with a fuzzy poset, we prove that there is a fuzzy ordering relation on the same set with equal cuts. Vice versa, for every fuzzy ordering relation there is a fuzzy poset on the same underlying set, so that cut-posets of two structures coincide. We also present some p
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