A similarity-based generalization of fuzzy orderings preserving the classical axioms. (English) Zbl 1113.03333

Summary: Equivalence relations and orderings are key concepts of mathematics. For both types of relations, formulations within the framework of fuzzy relations have been proposed already in the early days of fuzzy set theory. While similarity (indistinguishability) relations have turned out to be very useful tools, e.g. for the interpretation of fuzzy partitions and fuzzy controllers, the utilization of fuzzy orderings is still lagging far behind, although there are a lot of possible applications, for instance, in fuzzy preference modeling and fuzzy control. The present paper is devoted to this missing link. After a brief motivation, we will critically analyze the existing approach to fuzzy orderings. In the main part, an alternative approach to fuzzy orderings, which also takes the strong connection to gradual similarity into account, is proposed and studied in detail, including several constructions and representation results.


03E72 Theory of fuzzy sets, etc.
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[1] DOI: 10.1016/S0020-0255(71)80005-1 · Zbl 0218.02058 · doi:10.1016/S0020-0255(71)80005-1
[2] Klawonn F., Mathware Soft Comput. 3 pp 197– (1995)
[3] DOI: 10.1016/0165-0114(85)90096-X · Zbl 0609.04002 · doi:10.1016/0165-0114(85)90096-X
[4] Hohle U., Proc. EUFIT9S 1 pp 358– (1993)
[5] De Baets B., J. Fuzzy Math. 5 pp 471– (1997)
[6] DOI: 10.1016/0165-0114(93)90272-J · Zbl 0785.93059 · doi:10.1016/0165-0114(93)90272-J
[7] DOI: 10.1016/0165-0114(93)90473-U · Zbl 1002.03530 · doi:10.1016/0165-0114(93)90473-U
[8] DOI: 10.1109/3477.552182 · doi:10.1109/3477.552182
[9] DOI: 10.1016/S0020-7373(79)80040-1 · Zbl 0403.68075 · doi:10.1016/S0020-7373(79)80040-1
[10] DOI: 10.1016/0165-0114(80)90060-3 · Zbl 0433.03013 · doi:10.1016/0165-0114(80)90060-3
[11] DOI: 10.1016/0165-0114(91)90048-U · Zbl 0725.04003 · doi:10.1016/0165-0114(91)90048-U
[12] DOI: 10.1016/0165-0114(91)90258-R · Zbl 0747.90006 · doi:10.1016/0165-0114(91)90258-R
[13] DOI: 10.1016/S0165-0114(96)00331-4 · Zbl 0930.03070 · doi:10.1016/S0165-0114(96)00331-4
[14] DOI: 10.1016/0165-0114(93)90141-4 · Zbl 0795.04007 · doi:10.1016/0165-0114(93)90141-4
[15] DOI: 10.1016/0165-0114(94)00308-T · Zbl 0852.04011 · doi:10.1016/0165-0114(94)00308-T
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