ScienceDirect - Information Sciences : Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory I

Information Sciences
Volume 177, Issue 4, 15 February 2007, Pages 1007-1026

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doi:10.1016/j.ins.2006.07.011

Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory I

Xiaodong Liu^a^,^b^,^c^,^,, Tianyou Chai^b, Wei Wang^c and Wanquan Liu^c

^aResearch Center of Information and Control, Dalian University of Technology, Dalian 116024, PR China ^bResearch Center of Automation, Northeastern University, Shenyang 110004, PR China ^cDepartment of Computing, Curtin University of Technology, Bentley, WA 6102, Australia

Received 11 September 2004;

revised 2 April 2006;

accepted 5 July 2006.

Available online 4 August 2006.

References and further reading may be available for this article. To view references and further reading you must purchase this article.

Abstract

In this paper, the representations of fuzzy concepts based on raw data have been investigated within the framework of AFS (Axiomatic Fuzzy Set) theory. First, a brief review of AFS theory is presented and a completely distributive lattice, the E^#I algebra, is proposed. Secondly, two kinds of E^#I algebra representations of fuzzy concepts are derived in detail. In order to represent the membership functions of fuzzy concepts in the interval [0, 1], the norm of AFS algebra is defined and studied. Finally, the relationships of various representations with their advantages and drawbacks are analyzed.

Keywords: AFS algebras; Completely distributive lattices; AFS structures; Preference relations; Sub-preference relations

Article Outline

1. Introduction
2. A review of the AFS theory 2.1. AFS algebras 2.2. AFS structures 2.3. EII, EIII algebra representation of fuzzy sets
3. E^#Iⁿ algebra representations of fuzzy concepts
4. The norms for the AFS algebras
5. Conclusion
Acknowledgements
Appendix A. The proof of Theorem 7
Appendix B. The proof of Theorem 8
References

Information Sciences
Volume 177, Issue 4, 15 February 2007, Pages 1007-1026

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