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Intuitionistic fuzzy filter theory of BL-algebras

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Abstract

In this paper, the intuitionistic fuzzy filter theory of BL-algebras is researched. The basic knowledge of BL-algebras and intuitionistic fuzzy sets is firstly reviewed. The notions of intuitionistic fuzzy filters, lattice filters, prime filters, Boolean filters, implicative filters, positive implicative filters, ultra filters and obstinate filters are introduced, respectively. Their important properties are investigated. In intuitionistic fuzzy sets, intuitionistic fuzzy filters, Boolean filters, ultra filters are proved to be equivalent to lattice filters, implicative filters, obstinate filters, respectively. Each intuitionistic fuzzy Boolean filter is an intuitionistic fuzzy positive implicative filter, but the converse may not be true in BL-algebras. The conditions under an intuitionistic fuzzy positive implicative filter being an intuitionistic fuzzy Boolean filter are constructed. Finally, the concepts of the intuitionistic fuzzy ultra and obstinate filters are introduced, and the intuitionistic fuzzy ultra filter is proved to be equivalent to the intuitionistic fuzzy obstinate filter in BL-algebras.

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Acknowledgments

This work is supported by the national natural science foundation of China under Grant No. 61273018 and the natural science research project of the Education Department of Henan Province of China under Grant No. 2009A520015.

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Authors and Affiliations

  1. College of Computer and Information Technology, Henan Normal University, Xinxiang, China

    Zhan’ao Xue, Yunhua Xiao, Huiru Cheng & Yuejun Li

  2. Information and Communication Engineering Institute, North University of China, Taiyuan, China

    Weihua Liu

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  1. Zhan’ao Xue
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  2. Yunhua Xiao
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  3. Weihua Liu
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  4. Huiru Cheng
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  5. Yuejun Li
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Correspondence to Zhan’ao Xue.

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Xue, Z., Xiao, Y., Liu, W. et al. Intuitionistic fuzzy filter theory of BL-algebras. Int. J. Mach. Learn. & Cyber. 4, 659–669 (2013). https://doi.org/10.1007/s13042-012-0130-8

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  • DOI: https://doi.org/10.1007/s13042-012-0130-8

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