Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems

Abstract

The purpose of this paper is two-fold. Firstly, a general concept of closedness of fuzzy sets under fuzzy preorderings is proposed and investigated along with the corresponding opening and closure operators. Secondly, the practical impact of this notion is demonstrated by applying it to the analysis of ordering-based modifiers.

Keywords:

• Closedness
• Fuzzy preordering
• Fuzzy relation
• Linguistic modifier

Notes

^†This work was supported by the K ^plus Competence Center Program which is funded by the Austrian Government, the Province of Upper Austria, and the Chamber of Commerce of Upper Austria.

^‡This work was supported by the Fund for Scientific Research of Flanders (FWO). Parts of this work were done during a COST Action 15 short-term scientific mission of Martine De Cock at the Fuzzy Logic Laboratory Linz-Hagenberg in April 2000.

¹It depends on the underlying inference scheme whether the result is actually the same. We leave this aspect aside for the present paper, since this is not its major concern.

Ulrich Bodenhofer was born in Upper Austria in 1972. He received an M.Sc. degree in applied mathematics in 1996 and a PhD in technical sciences in 1998, both from the Johannes Kepler University, Linz, Austria. Dr Bodenhofer is currently scientific director of the Software Competence Center Hagenberg (SCCH). He is also working as a lecturer at the Johannes Kepler University, Linz. He is a member of IEEE, the IEEE Communications Society, the IEEE Systems, Man and Cybernetics Society, the IEEE Neural Networks Society, the North-American Fuzzy Information Processing Society (NAFIPS), and a board member of the European Society for Fuzzy Logic and Technology (EUSFLAT). His research interests include fuzzy and many-valued logics, fuzzy relations, decision analysis, genetic algorithms, machine learning, and image processing.

Martine De Cock was born on September 9, 1976. After obtaining an M.Sc. degree in Computer Science and a teaching degree at Ghent University in 1998, she became a member of the Fuzziness and Uncertainty Modelling Research Unit. In 2002, she obtained a PhD in Computer Science. Currently she is working as a postdoctoral researcher of the Fund for Scientific Research—Flanders (FWO).

Etienne E. Kerre was born in Zele, Belgium on May 8, 1945. He obtained his M.Sc. Degree in Mathematics in 1967 and his PhD in Mathematics in 1970 from Ghent University. Since 1984, he has been a lecturer, and since 1991, a full professor at Ghent University. He is a referee for more than 30 international scientific journals, and also a member of the editorial board of international journals and conferences on fuzzy set theory. He was an honorary chairman at various international conferences. In 1976, he founded the Fuzziness and Uncertainty Modelling Research Unit (FUM) and since then his research has been focused on the modelling of fuzziness and uncertainty, and has resulted in a great number of contributions in fuzzy set theory and its various generalizations, and in evidence theory. Especially the theories of fuzzy relational calculus and of fuzzy mathematical structures owe a very great deal to him. Over the years he has also been a promotor of 16 PhDs on fuzzy set theory. His current research interests include fuzzy and intuitionistic fuzzy relations, fuzzy topology, and fuzzy image processing. He has authored or co-authored 11 books, and more than 100 papers that have appeared in international refereed journals.