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Interval-valued intuitionistic (T, S)-fuzzy filters theory on residuated lattices

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Abstract

The aim of this paper is further to develop the filter theory on residuated lattices. Firstly, the notion of interval valued intuitionistic (T, S)-fuzzy filter(IVI (T, S)-fuzzy filter for short) on residuated lattices is introduced by linking the interval valued intuitionistic fuzzy set, t-norm, s-norm and filter theory of residuated lattices; the properties and equivalent characterizations of interval valued intuitionistic (T, S)-fuzzy filter are investigated; the relation between IVI (T, S)-fuzzy filter and filter is studied. Secondly, the notions of interval valued intuitionistic (T, S)-fuzzy implicative filter and interval valued intuitionistic (T, S)-fuzzy Boolean filter are introduced; the properties and equivalent characterizations of them are investigated; the intuitionistic (T, S)-fuzzy implicative filter is proved to be equivalent to the intuitionistic (T, S)-fuzzy Boolean filter in residuated lattices. Finally, the intuitionistic (T, S)-fuzzy positive implicative filter and intuitionistic (T, S)-fuzzy G (MV) filter are introduced; some equivalent characterizations of them are obtained and the relations among these fuzzy filters are investigated.

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Acknowledgments

We would like to thank the anonymous reviewers’ comments and suggestions improved both content and the presentation of this paper. One of reviewer point out many typing mistakes and grammar mistakes in the manuscript, we gave him (her) heartfelt thanks. This work was supported by National Natural Science Foundation of P.R.China (Grant No. 61175055), Sichuan Key Technology Research and Development Program (Grant No.2011FZ0051), Radio Administration Bureau of MIIT of China(Grant No.[2011]146), China Institution of Communications(Grant No.[2011]051). The Speciality Comprehensive Reform of Mathematics and Applied Mathematics of Ministry of Education(ZG0464). The Speciality Comprehensive Reform of Mathematics and Applied Mathematics of Ministry of Education(01249). A Project Supported by Scientific Research Fund of Sichuan Provincial Education Department. The Scientific Research Research Fund of Neijiang Normal University(No.13ZB05)

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Authors and Affiliations

  1. College of Mathematics and Information Sciences, Neijiang Normal University, Neijiang, 641000, Sichuan, People’s Republic of China

    Yi Liu

  2. College of Mathematics and Computer Sciences, Shanxi Norm University, Linfen, 041000, Shanxi, People’s Republic of China

    Xiaoyan Qin

  3. Intelligent Control Development Center, Southwest Jiaotong University, Chengdu, 610031, Sichuan, People’s Republic of China

    Yi Liu & Yang Xu

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  1. Yi Liu
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  2. Xiaoyan Qin
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  3. Yang Xu
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Correspondence to Yi Liu.

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Liu, Y., Qin, X. & Xu, Y. Interval-valued intuitionistic (T, S)-fuzzy filters theory on residuated lattices. Int. J. Mach. Learn. & Cyber. 5, 683–696 (2014). https://doi.org/10.1007/s13042-013-0213-1

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