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On Tarski’s Type Theorems for L-Fuzzy Isotone and L-Fuzzy Relatively Isotone Maps on L-Complete Propelattices
Abstract:Recently a new type of very general relational structures, the so called (L-)complete propelattices, was introduced. These significantly generalize complete lattices and completely lattice L-ordered sets, because they do not assume the technically very strong property of transitivity. For these structures also the main part of the original Tarski’s fixed point theorem holds for (L-fuzzy) isotone maps, i.e., the part which concerns the existence of fixed points and the structure of their set. In this paper, fundamental properties of (L-)complete propelattices are recalled and the so called L-fuzzy relatively isotone maps are introduced. For these maps it is proved that they also have fixed points in L-complete propelattices, even if their set does not have to be of an awaited analogous structure of a complete propelattice.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1126021Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 772
 F. Vcelar, Z. Patikova, On fuzzification of Tarskis fixed point theorem without transitivity (submitted to Fuzzy Sets and Systems).
 A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics 5 (1955), 285–309.
 J. Klimes, Fixed point characterization of completeness on lattices for relatively isotone mappings, Archivum Mathematicum 20 (3) (1984), 125–132.
 R. Belohlavek, Fuzzy Relational Systems: Foundations and Principles, Kluwer, New York, 2002.
 V. Novak, Fuzzy Sets and Their Applications, Adam-Hilger, Bristol, 1989.
 F. Vcelar, On an approach to the fuzzification of classical Arrow‘s aggregation problem, Int. J. of General Systems 2 (1994), 139–154.
 R. Belohlavek, Lattice-type fuzzy order is uniquely given by its 1-cut: proof and consequences, Fuzzy Sets and Systems 123 (2004), 447–458.
 P. Martinek, On generalization of fuzzy concept lattices based on change of underlying fuzzy order, CLA 2008, Proc. of the Sixth International Conference on Concept Lattices and Their Applications, Palacky University, Olomouc (2008), 207–215.
 E. Fried, G. Gr¨atzer, Partial and free weakly associative lattices, Houston J. of Mathematics 2 (4) (1976), 501–512.